Position of Inequality Or POI Metrics
Position Of Inequality or POI metrics
------
also called
Hawaiian Metrics or HOBBit metrics
======
Any time there are weighted categorical bands
(There are always "categorical" bSQ bit bands
after fully decomposing)
Rank order them in decreasing order of weight
E.g.,
Hobbit dist. = Max Position Of Inequality (MaxPOI)
Manhattan = Sums Weights of Pos Of Ineq (SumPOI or L1-POI)
Euclidean = SQRT(Sum SQRs of Wts of Pos Of Ineq) (L2-POI)
Minkowski-q = q-RT(Sum (Wts of Pos Of Ineq)^q) (Lq-POI)
Maximum = ??? Hobbit
These are the Hawaiian Metics.
======
When there are more than 2 categories (more than just 0 or 1),
1. Attributize the categories (ala MBR) or code them numeric?
2. Use Hawaiian Distance?
How might we relieve some of the "problems" of HMs?
- eccentricity of Hobbit rings?
- is it a problem? (see below under hobbit rings)
- thickness of Hobbit rings?
- is it a problem? (see below under hobbit rings)
assuming both are problems, how can we relieve them:
FIBONACCI HAWAIIAN METRICS and HOBBIT RINGS
======
If we think of binary (and decimal) digital coding of a number:
Start with binary base sequence, B = {..., 2^n, ..., 2^1, 2^0 }
(decimal base sequence, D = {..., 10^n, ...,10^1, 10^0}
Remove the largest base <= number (digit = # of copies removed)
Repeat with number := remainder until remainder = 0.
Code using Fibonacci sequence as base sequence (not B or D)
Fibonacci base sequence: ...233 144 89 55 34 21 13 8 5 3 2 1 1
( ni = n(i+1) + n(i+2) )
For byte data:
Index:13 12 11 10 9 8 7 6 5 4 3 2 1 0
Pos: 11 10 9 8 7 6 5 4 3 2 1 0
Fib: 233 144 89 55 34 21 13 8 5 3 2 1 _1__ 0
NUM seed
0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1
2 0 0 0 0 0 0 0 0 0 0 1 0
3 0 0 0 0 0 0 0 0 0 1 0 0
4 0 0 0 0 0 0 0 0 0 1 0 1
5 0 0 0 0 0 0 0 0 1 0 0 0
6 0 0 0 0 0 0 0 0 1 0 0 1
7 0 0 0 0 0 0 0 0 1 0 1 0
8 0 0 0 0 0 0 0 1 0 0 0 0
9 0 0 0 0 0 0 0 1 0 0 0 1
. . . (find the rest in the appendix)
More hobbit rings, thinner and better centered ;-))))
"Thin-ness of rings" needs to be studied and quantified.
To push the idea a little further, consider a Fibonacci starter
value of .1 rather than 1 (results in 16 bit representations and
results in more plateaus which should be even thinner ;-)))
159 98 61 37 23 14. 8.9 5.5 3.4 2.1 1.3 .8 .5 .3 .2 .1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0
2 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0
3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1
4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1
5 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0
6 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0
7 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0
8 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1
. . . (find the rest in the appendix)
Taking seed to be 1/B where B is any of Fibonacci {1,2,3,5,..}
gives a representation base which will always include 1
(include both copies of seed??).
******75 46 28. 17. 11 6.8 4.2 2.6 1.6 1 0.6 0.4 0.2 0.2
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1
2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1
3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1
4 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1
5 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1
6 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1
7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
. . . (find the rest in the appendix)
Fibonacci base provides multiple representers for most numbers.
For seed=s defines s-Canonical Fibonacci representation (sCF)
======
(when s=1 we will drop the s)
s-Packed Fibonacci (sPF) will be the representation with 1-bits
======moved as far right as possible, for s=1:
(see appendix for listing and CF-to-PF conversion)
Data Mining classification method based on Hawaiian Metrics
------
1. Form basic CFPtrees and basic PFPtrees (canonical and packed)
2. For a unclassified sample, x, form each hobbit ring mask as
CFring-i OR PFring-i = Hring-i
3. Apply Hring-i to both PFtrees and CFtrees
(OR the results together)
4. Vote weighting ratios should be according to the fibonacci
index of the ring (i.e., inner ring has index = 1, next ring
has index = 2 ...)
- This is a matter of using both represetations (CF and PF)
for both the sample and the training vectors.
- This will be more accurate than just applying the ORed
ring mask to CFtrees, but will be slower. That's the
tradeoff. "How much slower?" and "How much more accurate?"
General Note on Classification:
------
1. Most lazy classifiers (ours for example) may not be good
for finding edges! or separation boundaries (ala SVMs).
2. Most lazy classifiers are good for continuous classification
(Class Attribute is continuous or approximately so -
many classes, numeric)
- Most lazy classifiers are good at fitting, not edge
detection because they look for "fit" based on
continuity, not sudden change (edges).
3. It's amazing we did so well on the KDDcup02 since we used
a Lazy classifier (sort of) and the Class attribute was
definitely not continuous.
4. SVMs are good at edge detection
- finding support vectors is like finding the glow lines
of boundary between two classes (edges of the classes)
5. A new direction for binary or non-continuous classification:
Use podium-type approaches to find "support vectors"
======
- first, note, at least so far, we don't have a killer
improvement for SVMs (we only apply SVM methods
to classify. We don't have a killer idea
to either improve the speed of SVM classification,
improve the accuracy of SVM classification or both).
- Instead of using Podiums to manage training vector voting,
try to use hawaiian metrics to find all 1-dimensional
"support rings" (Really 1-D rings are intervals so
"support intervals". Looking for the first SI around
the sample where the class changes in that dimension).
- This should give us a collection of "support intervals"
which can then be "separation curve fitted" (either
in training space or in higher dimensional feature
space -after the SVM transformation is applied).
APPENDIX:
======
Fibonacci base provides multiple representers for most numbers.
For seed=s defines s-Canonical Fibonacci representation (sCF)
======
(when s=1 we will drop the s)
s-Packed Fibonacci (sPF) will be the representation with 1-bits
======moved as far right as possible, for s=1:
Fib: 233 144 89 55 34 21 13 8 5 3 2 1 < - seed
NUM
0 0 0 0 0 0 0 0 0 0 0 0 0 < - 1 rep
1 0 0 0 0 0 0 0 0 0 0 0 1 < - 1 rep
2 0 0 0 0 0 0 0 0 0 0 1 0 < - 1 rep
3 0 0 0 0 0 0 0 0 0 0 1 1
4 0 0 0 0 0 0 0 0 0 1 0 1 < - 1 rep
5 0 0 0 0 0 0 0 0 0 1 1 0 =(x+1)base2
6 0 0 0 0 0 0 0 0 0 1 1 1 =(x+1)base2
7 0 0 0 0 0 0 0 0 1 0 1 0 < - 1 rep
8 0 0 0 0 0 0 0 0 1 0 1 1 =(x+3)base2
9 0 0 0 0 0 0 0 0 1 1 0 1 =(x+4)base2
10 0 0 0 0 0 0 0 0 1 1 1 0 =(x+4)base2
11 0 0 0 0 0 0 0 1 0 0 1 1 =(x+8)base2
12 0 0 0 0 0 0 0 1 0 1 0 1 < - 1 rep
13 0 0 0 0 0 0 0 1 0 1 1 0 =(x+9)base2
14 0 0 0 0 0 0 0 1 0 1 1 1
15 0 0 0 0 0 0 0 1 1 0 1 0
16 0 0 0 0 0 0 0 1 1 0 1 1
17 0 0 0 0 0 0 0 1 1 1 0 1
18 0 0 0 0 0 0 1 0 0 1 1 0
19 0 0 0 0 0 0 1 0 0 1 1 1
20 0 0 0 0 0 0 1 0 1 0 1 0 < - 1 rep
21 0 0 0 0 0 0 1 0 1 0 1 1
22 0 0 0 0 0 0 1 0 1 1 0 1
23 0 0 0 0 0 0 1 0 1 1 1 0
24 0 0 0 0 0 0 1 0 1 1 1 1
25 0 0 0 0 0 0 1 1 0 1 0 1
26 0 0 0 0 0 0 1 1 0 1 1 0
27 0 0 0 0 0 0 1 1 0 1 1 1
28 0 0 0 0 0 0 1 1 1 0 1 0
29 0 0 0 0 0 0 1 1 1 0 1 1
30 0 0 0 0 0 0 1 1 1 1 0 1
31 0 0 0 0 0 0 1 1 1 1 1 0
32 0 0 0 0 0 0 1 1 1 1 1 1
33 0 0 0 0 0 1 0 1 0 1 0 1 < - 1 rep
34 0 0 0 0 0 1 0 1 0 1 1 0
. . . (find the rest further down in the appendix)
A 1-pass r-to-l alg which produces Binary PF from Binary CF
using table lookup:
- Forevery 1000 0000 0000 convert to 0101 0101 0110
- Forevery 100 0000 0000 convert to 010 1010 1011
- Forevery 10 0000 0000 convert to 01 0101 0110
- Forevery 1 0000 0000 convert to 0 1010 1011
- Forevery 1000 0000 convert to 0101 0110
- Forevery 100 0000 convert to 010 1011
- Forevery 10 0000 convert to 01 0110
- Forevery 1 0000 convert to 0 1011
- Forevery 1000 convert to 0110
- Forevery 100 convert to 011
- Forevery 1000 0000 0000 convert to 0101010101 10
- Forevery 100 0000 0000 convert to 0101010101 1
- Forevery 10 0000 0000 convert to 01010101 10
- Forevery 1 0000 0000 convert to 01010101 1
- Forevery 1000 0000 convert to 010101 10
- Forevery 100 0000 convert to 010101 1
- Forevery 10 0000 convert to 0101 10
- Forevery 1 0000 convert to 0101 1
- Forevery 1000 convert to 01 10
- Forevery 100 convert to 01 1
- In fact it would probably be efficient code to form the 256
entry table of all conversions.
- What we see from above is that BPF converted segments are
always alternating except for the last 1 or two digits.
Closed form formula for CFn:
Roots of E = r^2 = r + 1 are (1+-SRQ(5))/2
phi = (SQR(5)+1)/2 = -root2(E) ~= .618034...
Phi = 1 + phi = 1/phi = (SQR(5)+1)/2 = root1(E) ~= 1.618034...
CFn= ( (1+SQRT(5)/2)^n - (1-SQRT(5)/2)^n ) ( Phi^n - (-phi)^n)
Lim(n->inf)CFn/CFn+1 = Phi = ~1.61803... = gm (golden mean)
CFn alternates above and below gm.
Closed form formula for sCFn?
Closed form formula for the binary CFn representation?
Closed form formula for the binary sCFn representation?
Closed form formula for the binary PFn representation?
Closed form formula for the binary sPFn representation?
A rectangle with aspect ratio = gm has the nice recursive
(fractal?) property: Removing a maximal square leaves a
recatangle with aspect ratio = gm
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Given any Fibonacci sequence, {bn, bn-1,..., b1, b0}
where b0 is the seed, b1=b0+0, bn=bn-1+bn-2 for n>1
the canonical CF-b0 representation of a positive integer, x, is
QnQn-1...Q1Q0, where the Qi's are generated recursvely by:
Ri=x initially, for i = n, n-1, ... , 1, 0
IF bi <= Ri THEN Qi=1 and Ri-1 = Ri - bi ELSE Qi=0.
____Qi___
bi | Ri
Qi*bi
-----
Ri-1
This is taking out the maximum each time (left to right)
======
6. What about taking out the minimum (right to left?
Ri=x initially, for i = 0, 1, 2, ... , n
IF bi <= Ri THEN Qi=1 and Ri+1 = Ri - bi ELSE Qi=0.
Fibonacci base sequence: ...233 144 89 55 34 21 13 8 5 3 2 1 1
( ni = n(i+1) + n(i+2) )
For byte data:
Fibonacci seed
(use twice?)
Fib:233 144 89 55 34 21 13 8 5 3 2 1 1
Pos: 11 10 9 8 7 6 5 4 3 2 1 0 0
NUM:
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1 0
2 0 0 0 0 0 0 0 0 0 0 1 0 0
3 0 0 0 0 0 0 0 0 0 1 0 0 0
4 0 0 0 0 0 0 0 0 0 1 0 1 0
5 0 0 0 0 0 0 0 0 1 0 0 0 0
6 0 0 0 0 0 0 0 0 1 0 0 1
7 0 0 0 0 0 0 0 0 1 0 1 0
8 0 0 0 0 0 0 0 1 0 0 0 0
9 0 0 0 0 0 0 0 1 0 0 0 1
10 0 0 0 0 0 0 0 1 0 0 1 0
11 0 0 0 0 0 0 0 1 0 1 0 0
12 0 0 0 0 0 0 0 1 0 1 0 1
13 0 0 0 0 0 0 1 0 0 0 0 0
14 0 0 0 0 0 0 1 0 0 0 0 1
15 0 0 0 0 0 0 1 0 0 0 1 0
16 0 0 0 0 0 0 1 0 0 1 0 0
17 0 0 0 0 0 0 1 0 0 1 0 1
18 0 0 0 0 0 0 1 0 1 0 0 0
19 0 0 0 0 0 0 1 0 1 0 0 1
20 0 0 0 0 0 0 1 0 1 0 1 0
21 0 0 0 0 0 1 0 0 0 0 0 0
22 0 0 0 0 0 1 0 0 0 0 0 1
23 0 0 0 0 0 1 0 0 0 0 1 0
24 0 0 0 0 0 1 0 0 0 1 0 0
25 0 0 0 0 0 1 0 0 0 1 0 1
26 0 0 0 0 0 1 0 0 1 0 0 0
27 0 0 0 0 0 1 0 0 1 0 0 1
28 0 0 0 0 0 1 0 0 1 0 1 0
29 0 0 0 0 0 1 0 1 0 0 0 0
30 0 0 0 0 0 1 0 1 0 0 0 1
31 0 0 0 0 0 1 0 1 0 0 1 0
32 0 0 0 0 0 1 0 1 0 1 0 0
33 0 0 0 0 0 1 0 1 0 1 0 1
34 0 0 0 0 1 0 0 0 0 0 0 0
35 0 0 0 0 1 0 0 0 0 0 0 1
36 0 0 0 0 1 0 0 0 0 0 1 0
37 0 0 0 0 1 0 0 0 0 1 0 0
38 0 0 0 0 1 0 0 0 0 1 0 1
39 0 0 0 0 1 0 0 0 1 0 0 0
40 0 0 0 0 1 0 0 0 1 0 0 1
41 0 0 0 0 1 0 0 0 1 0 1 0
42 0 0 0 0 1 0 0 1 0 0 0 0
43 0 0 0 0 1 0 0 1 0 0 0 1
44 0 0 0 0 1 0 0 1 0 0 1 0
45 0 0 0 0 1 0 0 1 0 1 0 0
46 0 0 0 0 1 0 0 1 0 1 0 1
47 0 0 0 0 1 0 1 0 0 0 0 0
48 0 0 0 0 1 0 1 0 0 0 0 1
49 0 0 0 0 1 0 1 0 0 0 1 0
50 0 0 0 0 1 0 1 0 0 1 0 0
51 0 0 0 0 1 0 1 0 0 1 0 1
52 0 0 0 0 1 0 1 0 1 0 0 0
53 0 0 0 0 1 0 1 0 1 0 0 1
54 0 0 0 0 1 0 1 0 1 0 1 0
55 0 0 0 1 0 0 0 0 0 0 0 0
56 0 0 0 1 0 0 0 0 0 0 0 1
57 0 0 0 1 0 0 0 0 0 0 1 0
58 0 0 0 1 0 0 0 0 0 1 0 0
59 0 0 0 1 0 0 0 0 0 1 0 1
60 0 0 0 1 0 0 0 0 1 0 0 0
61 0 0 0 1 0 0 0 0 1 0 0 1
62 0 0 0 1 0 0 0 0 1 0 1 0
63 0 0 0 1 0 0 0 1 0 0 0 0
64 0 0 0 1 0 0 0 1 0 0 0 1
65 0 0 0 1 0 0 0 1 0 0 1 0
66 0 0 0 1 0 0 0 1 0 1 0 0
67 0 0 0 1 0 0 0 1 0 1 0 1
68 0 0 0 1 0 0 1 0 0 0 0 0
69 0 0 0 1 0 0 1 0 0 0 0 1
70 0 0 0 1 0 0 1 0 0 0 1 0
71 0 0 0 1 0 0 1 0 0 1 0 0
72 0 0 0 1 0 0 1 0 0 1 0 1
73 0 0 0 1 0 0 1 0 1 0 0 0
74 0 0 0 1 0 0 1 0 1 0 0 1
75 0 0 0 1 0 0 1 0 1 0 1 0
76 0 0 0 1 0 1 0 0 0 0 0 0
77 0 0 0 1 0 1 0 0 0 0 0 1
78 0 0 0 1 0 1 0 0 0 0 1 0
79 0 0 0 1 0 1 0 0 0 1 0 0
80 0 0 0 1 0 1 0 0 0 1 0 1
81 0 0 0 1 0 1 0 0 1 0 0 0
82 0 0 0 1 0 1 0 0 1 0 0 1
83 0 0 0 1 0 1 0 0 1 0 1 0
84 0 0 0 1 0 1 0 1 0 0 0 0
85 0 0 0 1 0 1 0 1 0 0 0 1
86 0 0 0 1 0 1 0 1 0 0 1 0
87 0 0 0 1 0 1 0 1 0 1 0 0
88 0 0 0 1 0 1 0 1 0 1 0 1
89 0 0 1 0 0 0 0 0 0 0 0 0
90 0 0 1 0 0 0 0 0 0 0 0 1
91 0 0 1 0 0 0 0 0 0 0 1 0
92 0 0 1 0 0 0 0 0 0 1 0 0
93 0 0 1 0 0 0 0 0 0 1 0 1
94 0 0 1 0 0 0 0 0 1 0 0 0
95 0 0 1 0 0 0 0 0 1 0 0 1
96 0 0 1 0 0 0 0 0 1 0 1 0
97 0 0 1 0 0 0 0 1 0 0 0 0
98 0 0 1 0 0 0 0 1 0 0 0 1
99 0 0 1 0 0 0 0 1 0 0 1 0
100 0 0 1 0 0 0 0 1 0 1 0 0
101 0 0 1 0 0 0 0 1 0 1 0 1
102 0 0 1 0 0 0 1 0 0 0 0 0
103 0 0 1 0 0 0 1 0 0 0 0 1
104 0 0 1 0 0 0 1 0 0 0 1 0
105 0 0 1 0 0 0 1 0 0 1 0 0
106 0 0 1 0 0 0 1 0 0 1 0 1
107 0 0 1 0 0 0 1 0 1 0 0 0
108 0 0 1 0 0 0 1 0 1 0 0 1
109 0 0 1 0 0 0 1 0 1 0 1 0
110 0 0 1 0 0 1 0 0 0 0 0 0
111 0 0 1 0 0 1 0 0 0 0 0 1
112 0 0 1 0 0 1 0 0 0 0 1 0
113 0 0 1 0 0 1 0 0 0 1 0 0
114 0 0 1 0 0 1 0 0 0 1 0 1
115 0 0 1 0 0 1 0 0 1 0 0 0
116 0 0 1 0 0 1 0 0 1 0 0 1
117 0 0 1 0 0 1 0 0 1 0 1 0
118 0 0 1 0 0 1 0 1 0 0 0 0
119 0 0 1 0 0 1 0 1 0 0 0 1
120 0 0 1 0 0 1 0 1 0 0 1 0
121 0 0 1 0 0 1 0 1 0 1 0 0
122 0 0 1 0 0 1 0 1 0 1 0 1
123 0 0 1 0 1 0 0 0 0 0 0 0
124 0 0 1 0 1 0 0 0 0 0 0 1
125 0 0 1 0 1 0 0 0 0 0 1 0
126 0 0 1 0 1 0 0 0 0 1 0 0
127 0 0 1 0 1 0 0 0 0 1 0 1
128 0 0 1 0 1 0 0 0 1 0 0 0
129 0 0 1 0 1 0 0 0 1 0 0 1
130 0 0 1 0 1 0 0 0 1 0 1 0
131 0 0 1 0 1 0 0 1 0 0 0 0
132 0 0 1 0 1 0 0 1 0 0 0 1
133 0 0 1 0 1 0 0 1 0 0 1 0
134 0 0 1 0 1 0 0 1 0 1 0 0
135 0 0 1 0 1 0 0 1 0 1 0 1
136 0 0 1 0 1 0 1 0 0 0 0 0
137 0 0 1 0 1 0 1 0 0 0 0 1
138 0 0 1 0 1 0 1 0 0 0 1 0
139 0 0 1 0 1 0 1 0 0 1 0 0
140 0 0 1 0 1 0 1 0 0 1 0 1
141 0 0 1 0 1 0 1 0 1 0 0 0
142 0 0 1 0 1 0 1 0 1 0 0 1
143 0 0 1 0 1 0 1 0 1 0 1 0
144 0 1 0 0 0 0 0 0 0 0 0 0
145 0 1 0 0 0 0 0 0 0 0 0 1
146 0 1 0 0 0 0 0 0 0 0 1 0
147 0 1 0 0 0 0 0 0 0 1 0 0
148 0 1 0 0 0 0 0 0 0 1 0 1
149 0 1 0 0 0 0 0 0 1 0 0 0
150 0 1 0 0 0 0 0 0 1 0 0 1
151 0 1 0 0 0 0 0 0 1 0 1 0
152 0 1 0 0 0 0 0 1 0 0 0 0
153 0 1 0 0 0 0 0 1 0 0 0 1
154 0 1 0 0 0 0 0 1 0 0 1 0
155 0 1 0 0 0 0 0 1 0 1 0 0
156 0 1 0 0 0 0 0 1 0 1 0 1
157 0 1 0 0 0 0 1 0 0 0 0 0
158 0 1 0 0 0 0 1 0 0 0 0 1
159 0 1 0 0 0 0 1 0 0 0 1 0
160 0 1 0 0 0 0 1 0 0 1 0 0
161 0 1 0 0 0 0 1 0 0 1 0 1
162 0 1 0 0 0 0 1 0 1 0 0 0
163 0 1 0 0 0 0 1 0 1 0 0 1
164 0 1 0 0 0 0 1 0 1 0 1 0
165 0 1 0 0 0 1 0 0 0 0 0 0
166 0 1 0 0 0 1 0 0 0 0 0 1
167 0 1 0 0 0 1 0 0 0 0 1 0
168 0 1 0 0 0 1 0 0 0 1 0 0
169 0 1 0 0 0 1 0 0 0 1 0 1
170 0 1 0 0 0 1 0 0 1 0 0 0
171 0 1 0 0 0 1 0 0 1 0 0 1
172 0 1 0 0 0 1 0 0 1 0 1 0
173 0 1 0 0 0 1 0 1 0 0 0 0
174 0 1 0 0 0 1 0 1 0 0 0 1
175 0 1 0 0 0 1 0 1 0 0 1 0
176 0 1 0 0 0 1 0 1 0 1 0 0
177 0 1 0 0 0 1 0 1 0 1 0 1
178 0 1 0 0 1 0 0 0 0 0 0 0
179 0 1 0 0 1 0 0 0 0 0 0 1
180 0 1 0 0 1 0 0 0 0 0 1 0
181 0 1 0 0 1 0 0 0 0 1 0 0
182 0 1 0 0 1 0 0 0 0 1 0 1
183 0 1 0 0 1 0 0 0 1 0 0 0
184 0 1 0 0 1 0 0 0 1 0 0 1
185 0 1 0 0 1 0 0 0 1 0 1 0
186 0 1 0 0 1 0 0 1 0 0 0 0
187 0 1 0 0 1 0 0 1 0 0 0 1
188 0 1 0 0 1 0 0 1 0 0 1 0
189 0 1 0 0 1 0 0 1 0 1 0 0
190 0 1 0 0 1 0 0 1 0 1 0 1
191 0 1 0 0 1 0 1 0 0 0 0 0
192 0 1 0 0 1 0 1 0 0 0 0 1
193 0 1 0 0 1 0 1 0 0 0 1 0
194 0 1 0 0 1 0 1 0 0 1 0 0
195 0 1 0 0 1 0 1 0 0 1 0 1
196 0 1 0 0 1 0 1 0 1 0 0 0
197 0 1 0 0 1 0 1 0 1 0 0 1
198 0 1 0 0 1 0 1 0 1 0 1 0
199 0 1 0 1 0 0 0 0 0 0 0 0
200 0 1 0 1 0 0 0 0 0 0 0 1
201 0 1 0 1 0 0 0 0 0 0 1 0
202 0 1 0 1 0 0 0 0 0 1 0 0
203 0 1 0 1 0 0 0 0 0 1 0 1
204 0 1 0 1 0 0 0 0 1 0 0 0
205 0 1 0 1 0 0 0 0 1 0 0 1
206 0 1 0 1 0 0 0 0 1 0 1 0
207 0 1 0 1 0 0 0 1 0 0 0 0
208 0 1 0 1 0 0 0 1 0 0 0 1
209 0 1 0 1 0 0 0 1 0 0 1 0
210 0 1 0 1 0 0 0 1 0 1 0 0
211 0 1 0 1 0 0 0 1 0 1 0 1
212 0 1 0 1 0 0 1 0 0 0 0 0
213 0 1 0 1 0 0 1 0 0 0 0 1
214 0 1 0 1 0 0 1 0 0 0 1 0
215 0 1 0 1 0 0 1 0 0 1 0 0
216 0 1 0 1 0 0 1 0 0 1 0 1
217 0 1 0 1 0 0 1 0 1 0 0 0
218 0 1 0 1 0 0 1 0 1 0 0 1
219 0 1 0 1 0 0 1 0 1 0 1 0
220 0 1 0 1 0 1 0 0 0 0 0 0
221 0 1 0 1 0 1 0 0 0 0 0 1
222 0 1 0 1 0 1 0 0 0 0 1 0
223 0 1 0 1 0 1 0 0 0 1 0 0
224 0 1 0 1 0 1 0 0 0 1 0 1
225 0 1 0 1 0 1 0 0 1 0 0 0
226 0 1 0 1 0 1 0 0 1 0 0 1
227 0 1 0 1 0 1 0 0 1 0 1 0
228 0 1 0 1 0 1 0 1 0 0 0 0
229 0 1 0 1 0 1 0 1 0 0 0 1
230 0 1 0 1 0 1 0 1 0 0 1 0
231 0 1 0 1 0 1 0 1 0 1 0 0
232 0 1 0 1 0 1 0 1 0 1 0 1
233 1 0 0 0 0 0 0 0 0 0 0 0
234 1 0 0 0 0 0 0 0 0 0 0 1
235 1 0 0 0 0 0 0 0 0 0 1 0
236 1 0 0 0 0 0 0 0 0 1 0 0
237 1 0 0 0 0 0 0 0 0 1 0 1
238 1 0 0 0 0 0 0 0 1 0 0 0
239 1 0 0 0 0 0 0 0 1 0 0 1
240 1 0 0 0 0 0 0 0 1 0 1 0
241 1 0 0 0 0 0 0 1 0 0 0 0
242 1 0 0 0 0 0 0 1 0 0 0 1
243 1 0 0 0 0 0 0 1 0 0 1 0
244 1 0 0 0 0 0 0 1 0 1 0 0
245 1 0 0 0 0 0 0 1 0 1 0 1
246 1 0 0 0 0 0 1 0 0 0 0 0
247 1 0 0 0 0 0 1 0 0 0 0 1
248 1 0 0 0 0 0 1 0 0 0 1 0
249 1 0 0 0 0 0 1 0 0 1 0 0
250 1 0 0 0 0 0 1 0 0 1 0 1
251 1 0 0 0 0 0 1 0 1 0 0 0
252 1 0 0 0 0 0 1 0 1 0 0 1
253 1 0 0 0 0 0 1 0 1 0 1 0
254 1 0 0 0 0 1 0 0 0 0 0 0
255 1 0 0 0 0 1 0 0 0 0 0 1
More hobbit rings, thinner and better centered ;-))))
To push the idea a little further, consider a Fibonacci starter
value of .1 rather than 1 (results in 16 bit representations and
results in more plateaus which should be even thinner ;-)))
Fib 159 98 61 37 23 14. 8.9 5.5 3.4 2.1 1.3 0.8 0.5 0.3 0.2 0.1 0.1
Pos: 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1
2 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1
3 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0
4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
5 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0
6 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1
7 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1
8 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0
9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
10 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0
11 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0
12 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1
13 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1
14 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0
15 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
16 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0
17 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1
18 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1
19 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0
20 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
21 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0
22 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0
23 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1
24 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1
25 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
26 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0
27 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0
28 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1
29 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1
30 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0
31 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0
32 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0
33 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0
34 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1
35 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1
36 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0
37 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0
38 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0
39 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1
40 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
41 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0
42 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0
43 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0
44 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0
45 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1
46 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1
47 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
48 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0
49 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0
50 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1
51 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1
52 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1
53 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0
54 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0
55 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0
56 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1
57 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1
58 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0
59 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0
60 0 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0
61 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1
62 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1
63 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1
64 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0
65 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0
66 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0
67 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1
68 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1
69 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0
70 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0
71 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0
72 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0
73 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1
74 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1
75 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0
76 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0
77 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0
78 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1
79 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1
80 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0
81 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0
82 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0
83 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0
84 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1
85 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1
86 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0
87 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0
88 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0
89 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1
90 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1
91 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0
92 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0
93 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0
94 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0
95 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1
96 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1
97 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0
98 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0
99 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
100 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1
101 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1
102 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0
103 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0
104 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0
105 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0
106 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1
107 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1
108 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0
109 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0
110 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0
111 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1
112 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1
113 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1
114 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0
115 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0
116 0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0
117 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1
118 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1
119 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0
120 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0
121 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0
122 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1
123 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1
124 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1
125 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0
126 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0
127 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0
128 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1
129 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1
130 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0
131 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0
132 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0
133 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0
134 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1
135 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1
136 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0
137 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0
138 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0
139 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 1
140 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1
141 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0
142 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0
143 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0
144 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0
145 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1
146 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1
147 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0
148 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0
149 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0
150 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1
151 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1
152 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0
153 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0
154 0 1 0 1 0 1 0 0 0 1 0 1 0 0 1 0
155 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0
156 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1
157 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1
158 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 0
159 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0
160 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
161 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
162 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
163 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0
164 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0
165 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0
166 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0
167 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1
168 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1
169 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0
170 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
171 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0
172 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1
173 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1
174 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1
175 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
176 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0
177 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0
178 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1
179 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1
180 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
181 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0
182 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0
183 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1
184 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1
185 1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1
186 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0
187 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
188 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0
189 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1
190 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1
191 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0
192 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0
193 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0
194 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0
195 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
196 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1
197 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0
198 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
199 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
200 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1
201 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1
202 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0
203 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
204 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0
205 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0
206 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1
207 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1
208 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0
209 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
210 1 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0
211 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1
212 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1
213 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0
214 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0
215 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0
216 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0
217 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1
218 1 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1
219 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0
220 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0
221 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0
222 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1
223 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
224 1 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0
225 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0
226 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0
227 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0
228 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1
229 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1
230 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0
231 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0
232 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0
233 1 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1
234 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1
235 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1
236 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0
237 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0
238 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0
239 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1
240 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1
241 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0
242 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0
243 1 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0
244 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1
245 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1
246 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1
247 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0
248 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0
249 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0
250 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1
251 1 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1
252 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0
253 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0
254 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0
255 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0
"Thin-ness of plateaus" needs to be studied and quantified.
NOTES on FIBBONACCI NUMBERS and SEQUENCES:
1. Taking the fraction to be 1/B where B is any of
the standard Fibonacci numbers {1,2,3,5,8,13,21,34...}
gives a sequence of base numbers which will include 1.
******75 46 28. 17. 11 6.8 4.2 2.6 1.6 1 0.6 0.4 0.2 0.2 = 1/5
Pos:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1
2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1
3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1
4 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1
5 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1
6 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1
7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
8 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1
9 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1
10 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1
11 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1
12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1
13 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1
14 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1
15 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1
16 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1
17 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1
18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
19 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1
20 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1
21 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1
22 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1
23 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1
24 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1
25 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1
26 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1
27 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1
28 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1
29 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
30 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1
31 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1
32 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1
33 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1
34 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1
35 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1
36 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1
37 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1
38 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1
39 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1
40 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1
41 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1
42 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1
43 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1
44 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1
45 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1
46 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1
47 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1
48 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1
49 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1
50 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1
51 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1
52 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1
53 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1
54 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1
55 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1
56 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1
57 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1
58 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1
59 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1
60 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1
61 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
62 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1
63 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1
64 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1
65 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1
66 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1
67 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1
68 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 1
69 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1
70 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 1
71 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1
72 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1
73 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1
74 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1
75 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1
76 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1
77 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1
78 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1
79 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1
80 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1
81 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1
82 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1
83 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1
84 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1
85 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1
86 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1
87 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 1
88 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1
89 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1
90 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1
91 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1
92 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1
93 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1
94 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1
95 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1
96 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1
97 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1
98 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1
99 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1
100 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1
101 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1
102 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 1
103 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1
104 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1
105 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1
106 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1
107 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1
108 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1
109 0 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1
110 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1
111 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1
112 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1
113 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 1
114 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1
115 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 1
116 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1
117 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1
118 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1
119 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1
120 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1
121 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1
122 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1
123 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1
124 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1
125 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1
126 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1
127 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1
128 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1
129 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1
130 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1
131 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1
132 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1
133 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1
134 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1
135 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1
136 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1
137 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1
138 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1
139 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1
140 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1
141 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1
142 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 1
143 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1
144 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1
145 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 1
146 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1
147 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1
148 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 1
149 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1
150 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1
151 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1
152 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1
153 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1
154 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1
155 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1
156 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1
157 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1
158 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1
159 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1
160 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1
161 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1
162 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1
163 0 1 0 0 1 0 1 0 0 0 0 1 0 0 0 1
164 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1
165 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1
166 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 1
167 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1
168 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1
169 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1
170 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1
171 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1
172 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1
173 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1
174 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1
175 0 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1
176 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1
177 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1
178 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1
179 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1
180 0 1 0 1 0 0 1 0 0 0 0 0 0 0 1 1
181 0 1 0 1 0 0 1 0 0 0 0 1 0 0 1 1
182 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1
183 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1
184 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1
185 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 1
186 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1
187 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1
188 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1
189 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1
190 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1
191 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 1
192 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1
193 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1
194 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1
195 0 1 0 1 0 1 0 1 0 0 1 0 0 0 0 1
196 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1
197 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1
198 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
199 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1
200 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1
201 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1
202 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1
203 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1
204 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1
205 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1
206 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1
207 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1
208 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1
209 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1
210 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1
211 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1
212 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1
213 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1
214 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1
215 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 1
216 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1
217 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
218 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1
219 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1
220 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1
221 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1
222 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1
223 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1
224 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1
225 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 1
226 1 0 0 0 0 1 0 1 0 1 0 1 0 0 1 1
227 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1
228 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1
229 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1
230 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1
231 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1
232 1 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1
233 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1
234 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1
235 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1
236 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1
237 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1
238 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1
239 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1
240 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1
241 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
242 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1
243 1 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1
244 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1
245 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1
246 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1
247 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1
248 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1
249 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1
250 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1
251 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1
252 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
253 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1
254 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1
255 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1
2. In the canonical Fibonacci sequence,
Lim(n -> inf)Fn/Fn-1 = ~1.61803... =gm, the golden mean
(and the convergences is oscillatory above and below gm).
3. There is a closed form formula for Fn
(nth element of the canoical Fibonacci sequence):
Fn = ( (1+SQRT(5)/2)^n - (1-SQRT(5)/2)^n )
------
SQRT(5)
4. A rectangle with aspect ratio = gm has the nice recursive
(fractal?) property: Removing a maximal square leaves a
recatangle with aspect ratio = gm
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5. Given any Fibonacci base sequence (FBS), {bn, bn-1, ..., b1, b0}
where b0 is the seed, b1=b0+0, bn=bn-1+bn-2 for n>1
the canonical FBS(b0) representation of a positive integer, x, is
QnQn-1...Q1Q0, where the Qi's are generated recursvely by:
Ri=x initially, for i = n, n-1, ... , 1, 0
IF bi <= Ri THEN Qi=1 and Ri-1 = Ri - bi ELSE Qi=0.
____Qi___
bi | Ri
Qi*bi
-----
Ri-1
This is taking out the maximum each time (left to right)
======
6. What about taking out the minimum (right to left?
Ri=x initially, for i = 0, 1, 2, ... , n
IF bi <= Ri THEN Qi=1 and Ri+1 = Ri - bi ELSE Qi=0.
____Qi___
bi | Ri
Qi*bi
-----
Ri+1
Problems: Only an estimate of x is produce
and multiple x's can have the same representative.
That about taking a geometric sequence with base (~1.61803)^2 = gm^2 ?
************76. 46. 29. 17. 11. 6.8 4.2 2.6 1.6 1 0.6 0.3 .
Pos:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1/(gm)^2
num_
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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33 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
34 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0
35 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0
36 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0
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38 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1
39 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1
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52 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0
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54 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
55 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0
56 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1
57 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1
58 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1
59 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
60 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0
61 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0
62 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0
63 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0
64 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0
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66 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
67 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1
68 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1
69 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1
70 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0
71 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0
72 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0
73 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0
74 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 1
75 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1
76 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1
77 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
78 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
79 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
80 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0
81 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0
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98 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1
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100 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0
101 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0
102 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0
103 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1
104 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1
105 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1
106 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0
107 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0
108 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0
109 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0
110 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0
111 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0
112 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0
113 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0
114 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1
115 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1
116 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1
117 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0
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119 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0
120 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0
121 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0
122 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0
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125 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1
126 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1
127 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1
128 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
129 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0
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131 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0
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134 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1
135 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0
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143 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1
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147 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0
148 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0
149 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0
150 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1
151 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1
152 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1
153 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0
154 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0
155 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0
156 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0
157 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0
158 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0
159 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0
160 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0
161 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
162 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1
163 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1
164 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0
165 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0
166 0 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0
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171 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0
172 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1
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175 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0
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202 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1
203 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1
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211 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
212 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0
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214 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0
215 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0
216 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0
217 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0
218 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0
219 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1
220 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1
221 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1
222 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0
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224 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0
225 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0
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227 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1
228 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 1
229 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0
230 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0
231 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0
232 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0
233 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0
234 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0
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236 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0
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*************************************************************
CF
21
3485321
349541385321
0 000000000000
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178 010010000000
179 010010000001
180 010010000010
181 010010000100
182 010010000101
183 010010001000
184 010010001001
185 010010001010
186 010010010000
187 010010010001
188 010010010010
189 010010010100
190 010010010101
191 010010100000
192 010010100001
193 010010100010
194 010010100100
195 010010100101
196 010010101000
197 010010101001
198 010010101010
199 010100000000
200 010100000001
201 010100000010
202 010100000100
203 010100000101
204 010100001000
205 010100001001
206 010100001010
207 010100010000
208 010100010001
209 010100010010
210 010100010100
211 010100010101
212 010100100000
213 010100100001
214 010100100010
215 010100100100
216 010100100101
217 010100101000
218 010100101001
219 010100101010
220 010101000000
221 010101000001
222 010101000010
223 010101000100
224 010101000101
225 010101001000
226 010101001001
227 010101001010
228 010101010000
229 010101010001
230 010101010010
231 010101010100
232 010101010101
233 100000000000
234 100000000001
235 100000000010
236 100000000100
237 100000000101
238 100000001000
239 100000001001
240 100000001010
241 100000010000
242 100000010001
243 100000010010
244 100000010100
245 100000010101
246 100000100000
247 100000100001
248 100000100010
249 100000100100
250 100000100101
251 100000101000
252 100000101001
253 100000101010
254 100001000000
255 100001000001
*******************************
CF (leading zeros deleted)
0
1 1
2 10
3 100
4 101
5 1000
6 1001
7 1010
8 10000
9 10001
10 10010
11 10100
12 10101
13 100000
14 100001
15 100010
16 100100
17 100101
18 101000
19 101001
20 101010
21 1000000
22 1000001
23 1000010
24 1000100
25 1000101
26 1001000
27 1001001
28 1001010
29 1010000
30 1010001
31 1010010
32 1010100
33 1010101
34 10000000
35 10000001
36 10000010
37 10000100
38 10000101
39 10001000
40 10001001
41 10001010
42 10010000
43 10010001
44 10010010
45 10010100
46 10010101
47 10100000
48 10100001
49 10100010
50 10100100
51 10100101
52 10101000
53 10101001
54 10101010
55 100000000
56 100000001
57 100000010
58 100000100
59 100000101
60 100001000
61 100001001
62 100001010
63 100010000
64 100010001
65 100010010
66 100010100
67 100010101
68 100100000
69 100100001
70 100100010
71 100100100
72 100100101
73 100101000
74 100101001
75 100101010
76 101000000
77 101000001
78 101000010
79 101000100
80 101000101
81 101001000
82 101001001
83 101001010
84 101010000
85 101010001
86 101010010
87 101010100
88 101010101
89 1000000000
90 1000000001
91 1000000010
92 1000000100
93 1000000101
94 1000001000
95 1000001001
96 1000001010
97 1000010000
98 1000010001
99 1000010010
100 1000010100
101 1000010101
102 1000100000
103 1000100001
104 1000100010
105 1000100100
106 1000100101
107 1000101000
108 1000101001
109 1000101010
110 1001000000
111 1001000001
112 1001000010
113 1001000100
114 1001000101
115 1001001000
116 1001001001
117 1001001010
118 1001010000
119 1001010001
120 1001010010
121 1001010100
122 1001010101
123 1010000000
124 1010000001
125 1010000010
126 1010000100
127 1010000101
128 1010001000
129 1010001001
130 1010001010
131 1010010000
132 1010010001
133 1010010010
134 1010010100
135 1010010101
136 1010100000
137 1010100001
138 1010100010
139 1010100100
140 1010100101
141 1010101000
142 1010101001
143 1010101010
144 10000000000
145 10000000001
146 10000000010
147 10000000100
148 10000000101
149 10000001000
150 10000001001
151 10000001010
152 10000010000
153 10000010001
154 10000010010
155 10000010100
156 10000010101
157 10000100000
158 10000100001
159 10000100010
160 10000100100
161 10000100101
162 10000101000
163 10000101001
164 10000101010
165 10001000000
166 10001000001
167 10001000010
168 10001000100
169 10001000101
170 10001001000
171 10001001001
172 10001001010
173 10001010000
174 10001010001
175 10001010010
176 10001010100
177 10001010101
178 10010000000
179 10010000001
180 10010000010
181 10010000100
182 10010000101
183 10010001000
184 10010001001
185 10010001010
186 10010010000
187 10010010001
188 10010010010
189 10010010100
190 10010010101
191 10010100000
192 10010100001
193 10010100010
194 10010100100
195 10010100101
196 10010101000
197 10010101001
198 10010101010
199 10100000000
200 10100000001
201 10100000010
202 10100000100
203 10100000101
204 10100001000
205 10100001001
206 10100001010
207 10100010000
208 10100010001
209 10100010010
210 10100010100
211 10100010101
212 10100100000
213 10100100001
214 10100100010
215 10100100100
216 10100100101
217 10100101000
218 10100101001
219 10100101010
220 10101000000
221 10101000001
222 10101000010
223 10101000100
224 10101000101
225 10101001000
226 10101001001
227 10101001010
228 10101010000
229 10101010001
230 10101010010
231 10101010100
232 10101010101
233 100000000000
234 100000000001
235 100000000010
236 100000000100
237 100000000101
238 100000001000
239 100000001001
240 100000001010
241 100000010000
242 100000010001
243 100000010010
244 100000010100
245 100000010101
246 100000100000
247 100000100001
248 100000100010
249 100000100100
250 100000100101
251 100000101000
252 100000101001
253 100000101010
254 100001000000
255 100001000001
*************************************************************
PF (leading zeros included)
21
3485321
349541385321
0 000000000000
1 000000000001
2 000000000010
3 000000000011
4 000000000101
5 000000000110
6 000000000111
7 000000001010
8 000000001011
9 000000001101
10 000000001110
11 000000001111
12 000000010101
13 000000010110
14 000000010111
15 000000011010
16 000000011011
17 000000011101
18 000000011110
19 000000011111
20 000000101010
21 000000101011
22 000000101101
23 000000101110
24 000000101111
25 000000110101
26 000000110110
27 000000110111
28 000000111010
29 000000111011
30 000000111101
31 000000111110
32 000000111111
33 000001010101
34 000001010110
35 000001010111
36 000001011010
37 000001011011
38 000001011101
39 000001011110
40 000001011111
41 000001101010
42 000001101011
43 000001101101
44 000001101110
45 000001101111
46 000001110101
47 000001110110
48 000001110111
49 000001111010
50 000001111011
51 000001111101
52 000001111110
53 000001111111
54 000010101010
55 000010101010
56 000010101101
57 000010101110
58 000010101111
59 000010110101
60 000010110110
61 000010110111
62 000010111010
63 000010111011
64 000010111101
65 000010111110
66 000010111111
67 000011010101
68 000011010110
69 000011010111
70 000011011010
71 000011011011
72 000011011101
73 000011011110
74 000011011111
75 000011101010
76 000011101011
77 000011101101
78 000011101110
79 000011101111
80 000011110101
81 000011110110
82 000011110111
83 000011111010
84 000011111011
85 000011111101
86 000011111110
87 000011111111
88 000101010101
89 000101010110
90 000101010101
91 000101011010
92 000101011011
93 000101011101
94 000101011110
95 000101011111
96 000101101010
97 000101101011
98 000101101101
99 000101101110
011 000101101111
101 000101110101
102 000101110110
103 000101110111
104 000101111010
105 000101111011
106 000101111101
107 000101111110
108 000101111111
109 000110101010
110 000110101011
111 000110101101
112 000110101110
113 000110101111
114 000110110101
115 000110110110
116 000110110111
117 000110111010
118 000110111011
119 000110111101
120 000110111110
121 000110111111
122 000111010101
123 000111010110
124 000111010111
125 000111011010
126 000111011011
127 000111011101
128 000111011110
129 000111011111
130 000111101010
131 000111101011
132 000111101101
133 000111101110
134 000111101111
135 000111110101
136 000111110110
137 000111110111
138 000111111010
139 000111111011
140 000111111101
141 000111111110
142 000111111111
143 001010101010
144 001010101011
145 001010101101
146 001010101010
147 001010101111
148 001010110101
149 001010110110
150 001010110111
151 001010111010
152 001010111011
153 001010111101
154 001010111110
155 001010111111
156 001011010101
157 001011010110
158 001011010111
159 001011011010
160 001011011011
161 001011011101
162 001011011110
163 001011011111
164 001011101010
165 001011101011
166 001011101101
167 001011101110
168 001011101111
169 001011110101
170 001011110110
171 001011110111
172 001011111010
173 001011111011
174 001011111101
175 001011111110
176 001011111111
177 001101010101
178 001101010110
179 001101010111
180 001101011010
181 001101011011
182 001101011101
183 001101011110
184 001101011111
185 001101101010
186 001101101011
187 001101101101
188 001101101110
189 001101101111
190 001101110101
191 001101110110
192 001101110111
193 001101111010
194 001101111011
195 001101111101
196 001101111110
197 001101111111
198 001110101010
199 001110101010
200 001110101101
201 001110101110
202 001110101111
203 001110110101
204 001110110110
205 001110110111
206 001110111010
207 001110111011
208 001110111101
209 001110111110
210 001110111111
211 001111010101
212 001111010110
213 001111010111
214 001111011010
215 001111011011
216 001111011101
217 001111011110
218 001111011111
219 001111101010
220 001111101011
221 001111101101
222 001111101110
223 001111101111
224 001111110101
225 001111110110
226 001111110111
227 001111111010
228 001111111011
229 001111111101
230 001111111110
231 001111111111
232 010101010101
233 010101010110
234 010101010111
235 010101011010
236 010101011011
237 010101010101
238 010101011110
239 010101011111
240 010101101010
241 010101101011
242 010101101101
243 010101101110
244 010101101111
245 010101110101
246 010101110110
247 010101110111
248 010101111010
249 010101111011
250 010101111101
251 010101111110
252 010101111111
253 010110101010
254 010110101011
255 010110101101
***************************
PF (leading zeros deleted)
21
3485321
349541385321
vvvvvvvvvvvv
0
1 1
2 10
3 11
4 101
5 110
6 111
7 1010
8 1011
9 1101
10 1110
11 1111
12 10101
13 10110
14 10111
15 11010
16 11011
17 11101
18 11110
19 11111
20 101010
21 101011
22 101101
23 101110
24 101111
25 110101
26 110110
27 110111
28 111010
29 111011
30 111101
31 111110
32 111111
33 1010101
34 1010110
35 1010111
36 1011010
37 1011011
38 1011101
39 1011110
40 1011111
41 1101010
42 1101011
43 1101101
44 1101110
45 1101111
46 1110101
47 1110110
48 1110111
49 1111010
50 1111011
51 1111101
52 1111110
53 1111111
54 10101010
55 10101010
56 10101101
57 10101110
58 10101111
59 10110101
60 10110110
61 10110111
62 10111010
63 10111011
64 10111101
65 10111110
66 10111111
67 11010101
68 11010110
69 11010111
70 11011010
71 11011011
72 11011101
73 11011110
74 11011111
75 11101010
76 11101011
77 11101101
78 11101110
79 11101111
80 11110101
81 11110110
82 11110111
83 11111010
84 11111011
85 11111101
86 11111110
87 011111111
88 101010101
89 101010110
90 101010101
91 101011010
92 101011011
93 101011101
94 101011110
95 101011111
96 101101010
97 101101011
98 101101101
99 101101110
011 101101111
101 101110101
102 101110110
103 101110111
104 101111010
105 101111011
106 101111101
107 101111110
108 101111111
109 110101010
110 110101011
111 110101101
112 110101110
113 110101111
114 110110101
115 110110110
116 110110111
117 110111010
118 110111011
119 110111101
120 110111110
121 110111111
122 111010101
123 111010110
124 111010111
125 111011010
126 111011011
127 111011101
128 111011110
129 111011111
130 111101010
131 111101011
132 111101101
133 111101110
134 111101111
135 111110101
136 111110110
137 111110111
138 111111010
139 111111011
140 111111101
141 111111110
142 111111111
143 1010101010
144 1010101011
145 1010101101
146 1010101010
147 1010101111
148 1010110101
149 1010110110
150 1010110111
151 1010111010
152 1010111011
153 1010111101
154 1010111110
155 1010111111
156 1011010101
157 1011010110
158 1011010111
159 1011011010
160 1011011011
161 1011011101
162 1011011110
163 1011011111
164 1011101010
165 1011101011
166 1011101101
167 1011101110
168 1011101111
169 1011110101
170 1011110110
171 1011110111
172 1011111010
173 1011111011
174 1011111101
175 1011111110
176 1011111111
177 1101010101
178 1101010110
179 1101010111
180 1101011010
181 1101011011
182 1101011101
183 1101011110
184 1101011111
185 1101101010
186 1101101011
187 1101101101
188 1101101110
189 1101101111
190 1101110101
191 1101110110
192 1101110111
193 1101111010
194 1101111011
195 1101111101
196 1101111110
197 1101111111
198 1110101010
199 1110101010
200 1110101101
201 1110101110
202 1110101111
203 1110110101
204 1110110110
205 1110110111
206 1110111010
207 1110111011
208 1110111101
209 1110111110
210 1110111111
211 1111010101
212 1111010110
213 1111010111
214 1111011010
215 1111011011
216 1111011101
217 1111011110
218 1111011111
219 1111101010
220 1111101011
221 1111101101
222 1111101110
223 1111101111
224 1111110101
225 1111110110
226 1111110111
227 1111111010
228 1111111011
229 1111111101
230 1111111110
231 1111111111
232 10101010101
233 10101010110
234 10101010111
235 10101011010
236 10101011011
237 10101010101
238 10101011110
239 10101011111
240 10101101010
241 10101101011
242 10101101101
243 10101101110
244 10101101111
245 10101110101
246 10101110110
247 10101110111
248 10101111010
249 10101111011
250 10101111101
251 10101111110
252 10101111111
253 10110101010
254 10110101011
255 10110101101
**********************************
CF
PF
21
3485321
349541385321
vvvvvvvvvvvv
0
1 1
1 1
2 10
2 10
3 100
3 11
4 101
4 101
5 1000
5 110
6 1001
6 111
7 1010
7 1010
8 10000
8 1011
9 10001
9 1101
10 10010
10 1110
11 10100
11 1111
12 10101
12 10101
13 100000
13 10110
14 100001
14 10111
15 100010
15 11010
16 100100
16 11011
17 100101
17 11101
18 101000
18 11110
19 101001
19 11111
20 101010
20 101010
21 1000000
21 101011
22 1000001
22 101101
23 1000010
23 101110
24 1000100
24 101111
25 1000101
25 110101
26 1001000
26 110110
27 1001001
27 110111
28 1001010
28 111010
29 1010000
29 111011
30 1010001
30 111101
31 1010010
31 111110
32 1010100
32 111111
33 1010101
33 1010101
34 10000000
34 1010110
35 10000001
35 1010111
36 10000010
36 1011010
37 10000100
37 1011011
38 10000101
38 1011101
39 10001000
39 1011110
40 10001001
40 1011111
41 10001010
41 1101010
42 10010000
42 1101011
43 10010001
43 1101101
44 10010010
44 1101110
45 10010100
45 1101111
46 10010101
46 1110101
47 10100000
47 1110110
48 10100001
48 1110111
49 10100010
49 1111010
50 10100100
50 1111011
51 10100101
51 1111101
52 10101000
52 1111110
53 10101001
53 1111111
54 10101010
54 10101010
55 100000000
55 10101010
56 100000001
56 10101101
57 100000010
57 10101110
58 100000100
58 10101111
59 100000101
59 10110101
60 100001000
60 10110110
61 100001001
61 10110111
62 100001010
62 10111010
63 100010000
63 10111011
64 100010001
64 10111101
65 100010010
65 10111110
66 100010100
66 11001111
67 100010101
67 11010101
68 100100000
68 11010110
69 100100001
69 11010111
70 100100010
70 11011010
71 100100100
71 11011011
72 100100101
72 11011101
73 100101000
73 11011110
74 100101001
74 11011111
75 100101010
75 11101010
76 101000000
76 11101011
77 101000001
77 11101101
78 101000010
78 11101110
79 101000100
79 11101111
80 101000101
80 11110101
81 101001000
81 11110110
82 101001001
82 11110111
83 101001010
83 11111010
84 101010000
84 11111011
85 101010001
85 11111101
86 101010010
86 11111110
87 101010100
87 11111111
88 101010101
88 101010101
89 1000000000
89 101010110
90 1000000001
90 101010101
91 1000000010
91 101011010
92 1000000100
92 101011011
93 1000000101
93 101011101
94 1000001000
94 101011110
95 1000001001
95 101011111
96 1000001010
96 101101010
97 1000010000
97 101101011
98 1000010001
98 101101101
99 1000010010
99 101101110
100 1000010100
100 101101111
101 1000010101
101 101110101
102 1000100000
102 101110110
103 1000100001
103 101110111
104 1000100010
104 101111010
105 1000100100
105 101111011
106 1000100101
106 101111101
107 1000101000
107 101111110
108 1000101001
108 110011111
109 1000101010
109 110101010
110 1001000000
110 110101011
111 1001000001
111 110101101
112 1001000010
112 110101110
113 1001000100
113 110101111
114 1001000101
114 110110101
115 1001001000
115 110110110
116 1001001001
116 110110111
117 1001001010
117 110111010
118 1001010000
118 110111011
119 1001010001
119 110111101
120 1001010010
120 110111110
121 1001010100
121 111001111
122 1001010101
122 111010101
123 1010000000
123 111010110
124 1010000001
124 111010111
125 1010000010
125 111011010
126 1010000100
126 111011011
127 1010000101
127 111011101
128 1010001000
128 111011110
129 1010001001
129 111011111
130 1010001010
130 111101010
131 1010010000
131 111101011
132 1010010001
132 111101101
133 1010010010
133 111101110
134 1010010100
134 111101111
135 1010010101
135 111110101
136 1010100000
136 111110110
137 1010100001
137 111110111
138 1010100010
138 111111010
139 1010100100
139 111111011
140 1010100101
140 111111101
141 1010101000
141 111111110
142 1010101001
142 111111111
143 1010101010
143 1010101010
144 10000000000
144 1010101011
145 10000000001
145 1010101101
146 10000000010
146 1010101010
147 10000000100
147 1010101111
148 10000000101
148 1010110101
149 10000001000
149 1010110110
150 10000001001
150 1010110111
151 10000001010
151 1010111010
152 10000010000
152 1010111011
153 10000010001
153 1010111101
154 10000010010
154 1010111110
155 10000010100
155 1010111111
156 10000010101
156 1011010101
157 10000100000
157 1011010110
158 10000100001
158 1011010111
159 10000100010
159 1011011010
160 10000100100
160 1011011011
161 10000100101
161 1011011101
162 10000101000
162 1011011110
163 10000101001
163 1011011111
164 10000101010
164 1011101010
165 10001000000
165 1011101011
166 10001000001
166 1011101101
167 10001000010
167 1011101110
168 10001000100
168 1011101111
169 10001000101
169 1011110101
170 10001001000
170 1011110110
171 10001001001
171 1011110111
172 10001001010
172 1011111010
173 10001010000
173 1011111011
174 10001010001
174 1011111101
175 10001010010
175 1011111110
176 10001010100
176 1100111111
177 10001010101
177 1101010101
178 10010000000
178 1101010110
179 10010000001
179 1101010111
180 10010000010
180 1101011010
181 10010000100
181 1101011011
182 10010000101
182 1101011101
183 10010001000
183 1101011110
184 10010001001
184 1101011111
185 10010001010
185 1101101010
186 10010010000
186 1101101011
187 10010010001
187 1101101101
188 10010010010
188 1101101110
189 10010010100
189 1101101111
190 10010010101
190 1101110101
191 10010100000
191 1101110110
192 10010100001
192 1101110111
193 10010100010
193 1101111010
194 10010100100
194 1101111011
195 10010100101
195 1101111101
196 10010101000
196 1101111110
197 10010101001
197 1101111111
198 10010101010
198 1110101010
199 10100000000
199 1110101010
200 10100000001
200 1110101101
201 10100000010
201 1110101110
202 10100000100
202 1110101111
203 10100000101
203 1110110101
204 10100001000
204 1110110110
205 10100001001
205 1110110111
206 10100001010
206 1110111010
207 10100010000
207 1110111011
208 10100010001
208 1110111101
209 10100010010
209 1110111110
210 10100010100
210 1110111111
211 10100010101
211 1111010101
212 10100100000
212 1111010110
213 10100100001
213 1111010111
214 10100100010
214 1111011010
215 10100100100
215 1111011011
216 10100100101
216 1111011101
217 10100101000
217 1111011110
218 10100101001
218 1111011111
219 10100101010
219 1111101010
220 10101000000
220 1111101011
221 10101000001
221 1111101101
222 10101000010
222 1111101110
223 10101000100
223 1111101111
224 10101000101
224 1111110101
225 10101001000
225 1111110110
226 10101001001
226 1111110111
227 10101001010
227 1111111010
228 10101010000
228 1111111011
229 10101010001
229 1111111101
230 10101010010
230 1111111110
231 10101010100
231 10100111111
232 10101010101
232 10101010101
233 100000000000
233 10101010110
234 100000000001
234 10101010111
235 100000000010
235 10101011010
236 100000000100
236 10100111111
237 100000000101
237 10101010101
238 100000001000
238 10101011110
239 100000001001
239 10101011111
240 100000001010
240 10101101010
241 100000010000
241 10101101011
242 100000010001
242 10101101101
243 100000010010
243 10101101110
244 100000010100
244 10101101111
245 100000010101
245 10101110101
246 100000100000
246 10101110110
247 100000100001
247 10101110111
248 100000100010
248 10101111010
249 100000100100
249 10101111011
250 100000100101
250 10101111101
251 100000101000
251 10101111110
252 100000101001
252 10101111111
253 100000101010
253 10110101010
254 100001000000
254 10110101011
255 100001000001
255 10110101101
**********************************
CF PF
21 21
3485321 3485321
349541385321 349541385321
vvvvvvvvvvvv vvvvvvvvvvvv
0
1 1 1
2 10 10
3 100 11
4 101 101
5 1000 110
6 1001 111
7 1010 1010
8 10000 1011
9 10001 1101
10 10010 1110
11 10100 1111
12 10101 10101
13 100000 10110
14 100001 10111
15 100010 11010
16 100100 11011
17 100101 11101
18 101000 11110
19 101001 11111
20 101010 101010
21 1000000 101011
22 1000001 101101
23 1000010 101110
24 1000100 101111
25 1000101 110101
26 1001000 110110
27 1001001 110111
28 1001010 111010
29 1010000 111011
30 1010001 111101
31 1010010 111110
32 1010100 111111
33 1010101 1010101
34 10000000 1010110
35 10000001 1010111
36 10000010 1011010
37 10000100 1011011
38 10000101 1011101
39 10001000 1011110
40 10001001 1011111
41 10001010 1101010
42 10010000 1101011
43 10010001 1101101
44 10010010 1101110
45 10010100 1101111
46 10010101 1110101
47 10100000 1110110
48 10100001 1110111
49 10100010 1111010
50 10100100 1111011
51 10100101 1111101
52 10101000 1111110
53 10101001 1111111
54 10101010 10101010
55 100000000 10101010
56 100000001 10101101
57 100000010 10101110
58 100000100 10101111
59 100000101 10110101
60 100001000 10110110
61 100001001 10110111
62 100001010 10111010
63 100010000 10111011
64 100010001 10111101
65 100010010 10111110
66 100010100 11001111
67 100010101 11010101
68 100100000 11010110
69 100100001 11010111
70 100100010 11011010
71 100100100 11011011
72 100100101 11011101
73 100101000 11011110
74 100101001 11011111
75 100101010 11101010
76 101000000 11101011
77 101000001 11101101
78 101000010 11101110
79 101000100 11101111
80 101000101 11110101
81 101001000 11110110
82 101001001 11110111
83 101001010 11111010
84 101010000 11111011
85 101010001 11111101
86 101010010 11111110
87 101010100 11111111
88 101010101 101010101
89 1000000000 101010110
90 1000000001 101010101
91 1000000010 101011010
92 1000000100 101011011
93 1000000101 101011101
94 1000001000 101011110
95 1000001001 101011111
96 1000001010 101101010
97 1000010000 101101011
98 1000010001 101101101
99 1000010010 101101110
100 1000010100 101101111
101 1000010101 101110101
102 1000100000 101110110
103 1000100001 101110111
104 1000100010 101111010
105 1000100100 101111011
106 1000100101 101111101
107 1000101000 101111110
108 1000101001 110011111
109 1000101010 110101010
110 1001000000 110101011
111 1001000001 110101101
112 1001000010 110101110
113 1001000100 110101111
114 1001000101 110110101
115 1001001000 110110110
116 1001001001 110110111
117 1001001010 110111010
118 1001010000 110111011
119 1001010001 110111101
120 1001010010 110111110
121 1001010100 111001111
122 1001010101 111010101
123 1010000000 111010110
124 1010000001 111010111
125 1010000010 111011010
126 1010000100 111011011
127 1010000101 111011101
128 1010001000 111011110
129 1010001001 111011111
130 1010001010 111101010
131 1010010000 111101011
132 1010010001 111101101
133 1010010010 111101110
134 1010010100 111101111
135 1010010101 111110101
136 1010100000 111110110
137 1010100001 111110111
138 1010100010 111111010
139 1010100100 111111011
140 1010100101 111111101
141 1010101000 111111110
142 1010101001 111111111
143 1010101010 1010101010
144 10000000000 1010101011
145 10000000001 1010101101
146 10000000010 1010101010
147 10000000100 1010101111
148 10000000101 1010110101
149 10000001000 1010110110
150 10000001001 1010110111
151 10000001010 1010111010
152 10000010000 1010111011
153 10000010001 1010111101
154 10000010010 1010111110
155 10000010100 1010111111
156 10000010101 1011010101
157 10000100000 1011010110
158 10000100001 1011010111
159 10000100010 1011011010
160 10000100100 1011011011
161 10000100101 1011011101
162 10000101000 1011011110
163 10000101001 1011011111
164 10000101010 1011101010
165 10001000000 1011101011
166 10001000001 1011101101
167 10001000010 1011101110
168 10001000100 1011101111
169 10001000101 1011110101
170 10001001000 1011110110
171 10001001001 1011110111
172 10001001010 1011111010
173 10001010000 1011111011
174 10001010001 1011111101
175 10001010010 1011111110
176 10001010100 1100111111
177 10001010101 1101010101
178 10010000000 1101010110
179 10010000001 1101010111
180 10010000010 1101011010
181 10010000100 1101011011
182 10010000101 1101011101
183 10010001000 1101011110
184 10010001001 1101011111
185 10010001010 1101101010
186 10010010000 1101101011
187 10010010001 1101101101
188 10010010010 1101101110
189 10010010100 1101101111
190 10010010101 1101110101
191 10010100000 1101110110
192 10010100001 1101110111
193 10010100010 1101111010
194 10010100100 1101111011
195 10010100101 1101111101
196 10010101000 1101111110
197 10010101001 1101111111
198 10010101010 1110101010
199 10100000000 1110101010
200 10100000001 1110101101
201 10100000010 1110101110
202 10100000100 1110101111
203 10100000101 1110110101
204 10100001000 1110110110
205 10100001001 1110110111
206 10100001010 1110111010
207 10100010000 1110111011
208 10100010001 1110111101
209 10100010010 1110111110
210 10100010100 1110111111
211 10100010101 1111010101
212 10100100000 1111010110
213 10100100001 1111010111
214 10100100010 1111011010
215 10100100100 1111011011
216 10100100101 1111011101
217 10100101000 1111011110
218 10100101001 1111011111
219 10100101010 1111101010
220 10101000000 1111101011
221 10101000001 1111101101
222 10101000010 1111101110
223 10101000100 1111101111
224 10101000101 1111110101
225 10101001000 1111110110
226 10101001001 1111110111
227 10101001010 1111111010
228 10101010000 1111111011
229 10101010001 1111111101
230 10101010010 1111111110
231 10101010100 10100111111
232 10101010101 10101010101
233 100000000000 10101010110
234 100000000001 10101010111
235 100000000010 10101011010
236 100000000100 10100111111
237 100000000101 10101010101
238 100000001000 10101011110
239 100000001001 10101011111
240 100000001010 10101101010
241 100000010000 10101101011
242 100000010001 10101101101
243 100000010010 10101101110
244 100000010100 10101101111
245 100000010101 10101110101
246 100000100000 10101110110
247 100000100001 10101110111
248 100000100010 10101111010
249 100000100100 10101111011
250 100000100101 10101111101
251 100000101000 10101111110
252 100000101001 10101111111
253 100000101010 10110101010
254 100001000000 10110101011
255 100001000001 10110101101