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A097357
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For definition see Comments lines.
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5
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1, 2, 1, 3, 3, 4, 3, 6, 3, 6, 3, 7, 5, 8, 5, 11, 3, 6, 3, 9, 9, 12, 9, 16, 5, 10, 5, 13, 11, 16, 11, 22, 3, 6, 3, 9, 9, 12, 9, 18, 9, 18, 9, 21, 15, 24, 15, 31, 5, 10, 5, 15, 15, 20, 15, 28, 11, 22, 11, 27, 21, 32, 21, 43, 3, 6, 3, 9, 9, 12, 9, 18, 9, 18, 9, 21, 15, 24, 15, 33, 9, 18, 9, 27, 27
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Let b_n(m) represent the m-th entry of the n-th sequence (n > 0) of some family, with the following properties: (b_1(m)) = (0,1,0,0,0,0,0,0,0,0,...) (first term of sequence is m = 0 -> b_1(1)=1 ). For all m, n in naturals ( > 0 ):
Rule I: m > n > 0 -> b_n(m) = 0.
Rule II: b_n(n) = 1.
Rule III: |b_n(m+1) - b_n(m-1)| = 1 -> b_(n+1)(m) = 1 if b_n(m) = 0; b_(n+1)(m) = 0 if b_n(m) = 1; otherwise (i.e., |b_n(m+1) - b_n(m-1)| != 1 -> |b_n(m+1) - b_n(m-1)| = 0) b_(n+1)(m) = b_n(m).
Rule IV: b_n(0) = 0 (this is so that rule III can still be applied to b_n(1)).
The sequence (a(n)) = (a(1), a(2), ...) is then given by a(n) = Sum_{i>=0} b_n(i) = Sum_{i=1..n} b_n(i).
The sequence can be visualized as certain interactions between concentric rings.
This sequence may be connected with Sierpinski's triangle. Details of this as well as a visualization of the rules of "interaction" are given at the link. It is not currently known if this sequence is bounded. The various aligned "triangles of zeros" (apparently each with a number of rows equal to a factor of 8) one sees when using the computer program alude to Sierpinski's Triangle.
At certain points one notices that adjacent terms are all divisible by a certain number -- if this number is divided out one gets back initial terms of the sequence. For example, observe the subsequence (second line, above): 3,6,3,9,9,12,9,18,9,18,9,21,15,24,15,31,5,10,5,15,15,20,15,28,11,22,11,27, divide the first 15 terms by 3 -> 1,2,1,3,3,4,3,6,3,6,3,7,5,8,5 (this is the beginning of the sequence). Skip the number 31 and divide the next 7 terms by 5 -> (1,2,1,3,3,4,3). As the sequence gets longer, it apparently begins repeating (by some factor) an ever-increasing number of its initial terms; for example, another subsequence is: 3,6,3,9,9,12,9,18,9,18,9,21,15,24,15,33,9,18,9,27,27,36,27,48,15,30 = 3*(1,2,1,3,3,4,3,6,3,6,3,7,5,8,5,11,3,6,3,9,9,12,9,16,5,10).
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LINKS
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FORMULA
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EXAMPLE
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Table b_n(m), n >= 1, m >= 0, begins:
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 1, 0, 0, 0, 0, 0, 0, ...
0, 0, 1, 1, 1, 0, 0, 0, 0, 0, ...
0, 1, 0, 1, 0, 1, 0, 0, 0, 0, ...
0, 1, 0, 1, 0, 1, 1, 0, 0, 0, ...
0, 1, 0, 1, 0, 0, 0, 1, 0, 0, ...
0, 1, 0, 1, 1, 0, 1, 1, 1, 0, ...
See A128810 for another version. (End)
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PROG
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A simple Java program is given at the link provided.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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