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A296511 Number of toothpicks added at n-th stage to the toothpick structure of A296510.

%I A296511 #45 Nov 22 2022 11:56:13
%S A296511 1,2,4,6,6,6,6,10,16,20,16,10,6,10,16,24,28,32,28,32,40,50,40,22,8,10,
%T A296511 16,24,28,32,32,40,56,74,76,64,42,36,40,62,76,90,80,88,102,122,96,50,
%U A296511 14,10,16,24,28,32,32,40,56,74,76,64,46,44,56,82,104,124
%N A296511 Number of toothpicks added at n-th stage to the toothpick structure of A296510.
%C A296511 The structure and the behavior of this cellular automaton reveals that some cellular automata have recurrent periods that can be represented by irregular triangles of first differences whose row lengths are the terms of A011782 multiplied by k (instead of powers of 2), where k is the length of their "word". In this case the word must be "abc", therefore k = 3. In the case of the cellular automaton with normal toothpicks (A139250) the word must be "ab" and k = 2.
%C A296511 The associated sound to the animation of this cellular automaton could be [tick, tock, tack], [tic, tock, tack], and so on.
%C A296511 For more information about the "word" of a cellular automaton see A296612.
%H A296511 Rémy Sigrist, <a href="/A296511/b296511.txt">Table of n, a(n) for n = 1..6144</a>
%H A296511 Rémy Sigrist, <a href="/A296511/a296511.png">Illustration of the construction at generation 3*256</a>
%H A296511 Rémy Sigrist, <a href="/A296511/a296511.gp.txt">PARI program</a>
%H A296511 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A296511 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A296511 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e A296511 The structure of this irregular triangle is as shown below:
%e A296511    a, b, c;
%e A296511    a, b, c;
%e A296511    a, b, c, a, b, c;
%e A296511    a, b, c, a, b, c, a, b, c, a, b, c;
%e A296511    a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c;
%e A296511 ...
%e A296511 Every column is associated successively to one of the axes of the triangular grid.
%e A296511 Every row represents a geometric period of the cellular automaton.
%e A296511 So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 3, the sequence begins:
%e A296511    1, 2, 4;
%e A296511    6, 6, 6;
%e A296511    6,10,16,20,16,10;
%e A296511    6,10,16,24,28,32,28,32,40,50,40,22;
%e A296511    8,10,16,24,28,32,32,40,56,74,76,64,42,36,40,62,76,90,80,88,102,122,96,50;
%e A296511   14,10,16,24,28,32,32,40,56,74,76,64,...
%e A296511 ...
%o A296511 (PARI) See Links section.
%Y A296511 First differences of A296510.
%Y A296511 Cf. A160121 (word "a"), A139251 (word "ab"), A299477 (word "abcb"), A299479 (word "abcbc").
%Y A296511 Cf. A007283, A011782, A151906, A160121, A160161, A292612.
%K A296511 nonn,look,tabf
%O A296511 1,2
%A A296511 _Omar E. Pol_, Dec 14 2017
%E A296511 More terms from _Rémy Sigrist_, Jul 22 2022