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 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. 28
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS Table of n, a(n) for n=0..77. 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 FORMULA T(n,k) = k*A011782(n), with n >= 0 and k >= 1. EXAMPLE 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, ... CROSSREFS Cf. A011782, A147562, A147582, A139250, A139251, A160160, A160161, A296510, A296511, A296610, A296611, A299476, A299477, A299478, A299479. Sequence in context: A085201 A300401 A051601 * A193921 A074829 A060243 Adjacent sequences: A296609 A296610 A296611 * A296613 A296614 A296615 KEYWORD nonn,tabl AUTHOR Omar E. Pol, Jan 04 2018 STATUS approved

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