

A097357


For definition see Comments lines.


5



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
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OFFSET

1,2


COMMENTS

Let b_n(m) represent the mth entry of the nth 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(m1) = 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(m1) != 1 > b_n(m+1)b_n(m1) = 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...infty)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 ex., 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 ex., 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,)


LINKS

Table of n, a(n) for n=1..85.


FORMULA

a(n+1)=Sum(T(n,k) mod 2, 0<=k<=n) where T = A026300(Motzkin triangle), A064189, A084536, A091965, A110877, A125906, A126954 .  Philippe Deléham, Apr 28 2007


EXAMPLE

Comment from Philippe Deléham, Apr 28 2007: Table b_n(m) begins, n>=1, m>=0:
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 .


PROG

A simple java program is given at the link provided.


CROSSREFS

Sequence in context: A116921 A173989 A093068 * A123621 A151662 A049786
Adjacent sequences: A097354 A097355 A097356 * A097358 A097359 A097360


KEYWORD

nonn


AUTHOR

Creighton Dement, Aug 08 2004


STATUS

approved



