This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A097357 #16 Jan 03 2025 18:27:34
%S A097357 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,
%T A097357 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,
%U A097357 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
%N A097357 For definition see Comments lines.
%C A097357 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 ):
%C A097357 Rule I: m > n > 0 -> b_n(m) = 0.
%C A097357 Rule II: b_n(n) = 1.
%C A097357 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).
%C A097357 Rule IV: b_n(0) = 0 (this is so that rule III can still be applied to b_n(1)).
%C A097357 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).
%C A097357 The sequence can be visualized as certain interactions between concentric rings.
%C A097357 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.
%C A097357 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).
%F A097357 a(n+1) = Sum_{k=0..n} (T(n,k) mod 2) where T = A026300(Motzkin triangle), A064189, A084536, A091965, A110877, A125906, A126954. - _Philippe Deléham_, Apr 28 2007
%e A097357 From _Philippe Deléham_, Apr 28 2007: (Start)
%e A097357 Table b_n(m), n >= 1, m >= 0, begins:
%e A097357 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A097357 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
%e A097357 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, ...
%e A097357 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, ...
%e A097357 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, ...
%e A097357 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, ...
%e A097357 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, ...
%e A097357 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, ...
%e A097357 See A128810 for another version. (End)
%o A097357 (Java) // A simple Java program is given at the link provided.
%K A097357 nonn
%O A097357 1,2
%A A097357 _Creighton Dement_, Aug 08 2004