A097357 For definition see Comments lines.

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

Offset: 1

Author

Creighton Dement, Aug 08 2004

Keywords

• nonn

Comments

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).

Examples

			From _Philippe Deléham_, Apr 28 2007: (Start)
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)
		

Programs

Java

// A simple Java program is given at the link provided.

Formula

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