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

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, 16, 24, 28, 32, 32, 40, 56, 74, 76, 64, 42, 36, 40, 62, 76, 90, 80, 88, 102, 122, 96, 50, 14, 10, 16, 24, 28, 32, 32, 40, 56, 74, 76, 64, 46, 44, 56, 82, 104, 124

Offset

1,2

Comments

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.

The associated sound to the animation of this cellular automaton could be [tick, tock, tack], [tic, tock, tack], and so on.

For more information about the "word" of a cellular automaton see A296612.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..6144

Rémy Sigrist, Illustration of the construction at generation 3*256

Rémy Sigrist, PARI program

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

Index entries for sequences related to toothpick sequences

Example

The structure of this irregular triangle is as shown below:
   a, b, c;
   a, b, c;
   a, b, c, a, b, c;
   a, b, c, a, b, c, a, b, c, a, b, c;
   a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c;
...
Every column is associated successively to one of the axes of the triangular grid.
Every row represents a geometric period of the cellular automaton.
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 3, the sequence begins:
   1, 2, 4;
   6, 6, 6;
   6,10,16,20,16,10;
   6,10,16,24,28,32,28,32,40,50,40,22;
   8,10,16,24,28,32,32,40,56,74,76,64,42,36,40,62,76,90,80,88,102,122,96,50;
  14,10,16,24,28,32,32,40,56,74,76,64,...
...

Prog

(PARI) See Links section.

Crossrefs

First differences of A296510.

Cf. A160121 (word "a"), A139251 (word "ab"), A299477 (word "abcb"), A299479 (word "abcbc").

Cf. A007283, A011782, A151906, A160121, A160161, A292612.

Sequence in context: A287394 A302754 A225369 * A050823 A209863 A207540

Adjacent sequences: A296508 A296509 A296510 * A296512 A296513 A296514

Keyword

nonn,look,tabf

Author

Omar E. Pol, Dec 14 2017

Extensions

More terms from Rémy Sigrist, Jul 22 2022

Status

approved