A296611 -id:A296611 - cp's OEIS frontend

A296612 Square array read by antidiagonals upwards: T(n,k) equals k times the number of compositions (ordered partitions) of n, with n >= 0 and k >= 1.

1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 8, 8, 6, 4, 5, 16, 16, 12, 8, 5, 6, 32, 32, 24, 16, 10, 6, 7, 64, 64, 48, 32, 20, 12, 7, 8, 128, 128, 96, 64, 40, 24, 14, 8, 9, 256, 256, 192, 128, 80, 48, 28, 16, 9, 10, 512, 512, 384, 256, 160, 96, 56, 32, 18, 10, 11, 1024, 1024, 768, 512, 320, 192, 112, 64, 36, 20, 11, 12

Offset: 0

Omar E. Pol, Jan 04 2018

Also, at least for the first five columns, column k gives the row lengths of the irregular triangles of the first differences of the total number of elements in the structure of some cellular automata. Indeed, the study of the structure and the behavior of the toothpick cellular automaton on triangular grid (A296510), and other C.A. of the same family, reveals that some cellular automata that have recurrent periods 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 an internal cycle. This internal cycle is called here "word" of a cellular automaton (see examples).

			The corner of the square array begins:
    1,   2,   3,    4,    5,    6,    7,    8,    9,   10, ...
    1,   2,   3,    4,    5,    6,    7,    8,    9,   10, ...
    2,   4,   6,    8,   10,   12,   14,   16,   18,   20, ...
    4,   8,  12,   16,   20,   24,   28,   32,   36,   40, ...
    8,  16,  24,   32,   40,   48,   56,   64,   72,   80, ...
   16,  32,  48,   64,   80,   96,  112,  128,  144,  160, ...
   32,  64,  96,  128,  160,  192,  224,  256,  288,  320, ...
   64, 128, 192,  256,  320,  384,  448,  512,  576,  640, ...
  128, 256, 384,  512,  640,  768,  896, 1024, 1152, 1280, ...
  256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, ...
...
For k = 1 consider A160120, the Y-toothpick cellular automaton, which has word "a", so the structure of the irregular triangle of the first differences (A160161) is as follows:
a;
a;
a,a;
a,a,a,a;
a,a,a,a,a,a,a,a;
...
An associated sound to the animation of this cellular automaton could be (tick), (tick), (tick), ...
The row lengths of the above triangle are the terms of A011782, equaling the column 1 of the square array: 1, 1, 2, 4, 8, ...
.
For k = 2 consider A139250, the normal toothpick C.A. which has word "ab", so the structure of the irregular triangle of the first differences (A139251) is as follows:
a,b;
a,b;
a,b,a,b;
a,b,a,b,a,b,a,b;
a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
...
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
The row lengths of the above triangle are the terms of A011782 multiplied by 2, equaling the column 2 of the square array: 2, 2, 4, 8, 16, ...
.
For k = 3 consider A296510, the toothpicks C.A. on triangular grid, which has word "abc", so the structure of the irregular triangle of the first differences (A296511) is as follows:
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;
...
An associated sound to the animation could be (tick, tock, tack), (tick, tock, tack), ...
The row lengths of the above triangle are the terms of A011782 multiplied by 3, equaling the column 3 of the square array: 3, 3, 6, 12, 24, ...
.
For k = 4 consider A299476, the toothpick C.A. on triangular grid with word "abcb", so the structure of the irregular triangle of the first differences (A299477) is as follows:
a,b,c,b;
a,b,c,b;
a,b,c,b,a,b,c,b;
a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b;
a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b;
...
An associated sound to the animation could be (tick, tock, tack, tock), (tick, tock, tack, tock), ...
The row lengths of the above triangle are the terms of A011782 multiplied by 4, equaling the column 4 of the square array: 4, 4, 8, 16, 32, ...
.
For k = 5 consider A299478, the toothpick C.A. on triangular grid with word "abcbc", so the structure of the irregular triangle of the first differences (A299479) is as follows:
a,b,c,b,c;
a,b,c,b,c;
a,b,c,b,c,a,b,c,b,c;
a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c;
a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c;
...
An associated sound to the animation could be (tick, tock, tack, tock, tack), (tick, tock, tack, tock, tack), ...
The row lengths of the above triangle are the terms of A011782 multiplied by 5, equaling the column 5 of the square array: 5, 5, 10, 20, 40, ...

Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.
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

Cf. A011782, A147562, A147582, A139250, A139251, A160160, A160161, A296510, A296511, A296610, A296611, A299476, A299477, A299478, A299479.

T(n,k) = k*A011782(n), with n >= 0 and k >= 1.

A296610 Toothpick sequence on triangular grid in an infinite 60-degree wedge (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 13, 15, 18, 21, 25, 31, 36, 38, 41, 44, 48, 54, 61, 67, 75, 80, 88, 100, 110, 113, 116, 119, 123, 129, 136, 142, 150, 157, 167, 183, 199, 210, 220, 225, 233, 245, 261, 276, 295, 306, 325, 351, 372, 378, 381, 384, 388, 394, 401, 407, 415, 422, 432, 448, 464, 475, 485, 492, 502, 518, 538, 559, 585

Offset: 0

Omar E. Pol, Mar 02 2019

nonn

The rules are the same as the rules of A296510 (the toothpick sequence on triangular grid) but here we are in a 60-degree wedge. For the position of the initial toothpicks see the example.
a(n) gives the total number of toothpicks in the structure after n-th stage.
A296611, the first differences, gives the number of toothpicks added at n-th stage.
The "word" of this cellular automaton is "abc", the same as the word of A296510. For more information about the word of cellular automata see A296612.

			Illustration of the 60-degree wedge of the triangular grid and the first three terms of the sequence:
.
           /\             /\             /\
          /  \           / /\           / /\
         /    \         / /  \         /_/_ \
        /      \       /      \       /      \
       /        \     /        \     /        \
      /          \   /          \   /          \
n:          0              1              2
a(n):       0              1              2
.
At stage 0 there are no toothpicks in the wedge, so a(0) = 0.
At stage 1 we add a toothpick of length 2, so a(1) = 1.
At stage 2 we add a toothpick in horizontal position, so a(2) = a(1) + 1 = 1 + 1 = 2. Note that in the structure there is a trapeze of area 5.
Then, at stage 3 we add a toothpick such that a equilateral triangle of area 1 appears in the wedge.
Then, at stage 4 we add a toothpick placed in the same position as the first toothpick.
And so on.

Cf. A139250, A296510, A296611, A296612.

cp's OEIS Frontend

A296612 Square array read by antidiagonals upwards: T(n,k) equals k times the number of compositions (ordered partitions) of n, with n >= 0 and k >= 1.

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