This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A296612 #63 Sep 03 2020 12:19:05 %S A296612 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, %T A296612 64,48,32,20,12,7,8,128,128,96,64,40,24,14,8,9,256,256,192,128,80,48, %U A296612 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 %N 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. %C A296612 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). %H A296612 Thomas Grubb and Frederick Rajasekaran, <a href="https://arxiv.org/abs/2009.00650">Set Partition Patterns and the Dimension Index</a>, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence. %H A296612 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 A296612 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a> %H A296612 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %F A296612 T(n,k) = k*A011782(n), with n >= 0 and k >= 1. %e A296612 The corner of the square array begins: %e A296612 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... %e A296612 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... %e A296612 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... %e A296612 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... %e A296612 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ... %e A296612 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, ... %e A296612 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, ... %e A296612 64, 128, 192, 256, 320, 384, 448, 512, 576, 640, ... %e A296612 128, 256, 384, 512, 640, 768, 896, 1024, 1152, 1280, ... %e A296612 256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, ... %e A296612 ... %e A296612 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: %e A296612 a; %e A296612 a; %e A296612 a,a; %e A296612 a,a,a,a; %e A296612 a,a,a,a,a,a,a,a; %e A296612 ... %e A296612 An associated sound to the animation of this cellular automaton could be (tick), (tick), (tick), ... %e A296612 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, ... %e A296612 . %e A296612 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: %e A296612 a,b; %e A296612 a,b; %e A296612 a,b,a,b; %e A296612 a,b,a,b,a,b,a,b; %e A296612 a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b; %e A296612 ... %e A296612 An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound. %e A296612 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, ... %e A296612 . %e A296612 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: %e A296612 a,b,c; %e A296612 a,b,c; %e A296612 a,b,c,a,b,c; %e A296612 a,b,c,a,b,c,a,b,c,a,b,c; %e A296612 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 A296612 ... %e A296612 An associated sound to the animation could be (tick, tock, tack), (tick, tock, tack), ... %e A296612 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, ... %e A296612 . %e A296612 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: %e A296612 a,b,c,b; %e A296612 a,b,c,b; %e A296612 a,b,c,b,a,b,c,b; %e A296612 a,b,c,b,a,b,c,b,a,b,c,b,a,b,c,b; %e A296612 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; %e A296612 ... %e A296612 An associated sound to the animation could be (tick, tock, tack, tock), (tick, tock, tack, tock), ... %e A296612 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, ... %e A296612 . %e A296612 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: %e A296612 a,b,c,b,c; %e A296612 a,b,c,b,c; %e A296612 a,b,c,b,c,a,b,c,b,c; %e A296612 a,b,c,b,c,a,b,c,b,c,a,b,c,b,c,a,b,c,b,c; %e A296612 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; %e A296612 ... %e A296612 An associated sound to the animation could be (tick, tock, tack, tock, tack), (tick, tock, tack, tock, tack), ... %e A296612 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, ... %Y A296612 Cf. A011782, A147562, A147582, A139250, A139251, A160160, A160161, A296510, A296511, A296610, A296611, A299476, A299477, A299478, A299479. %K A296612 nonn,tabl %O A296612 0,3 %A A296612 _Omar E. Pol_, Jan 04 2018