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%I #20 Feb 24 2021 02:48:18
%S 1,1,2,1,3,4,1,4,7,5,1,5,10,11,7,1,6,13,19,15,10,1,7,16,29,25,23,13,1,
%T 8,19,41,37,40,35,14,1,9,22,55,51,61,67,43,16,1,10,25,71,67,86,109,94,
%U 47,19,1,11,28,89,85,115,161,173,100,55,22,1,12,31,109,105,148,223,286,181
%N Triangle read by rows in which the diagonals give the infinite set of Toothpick sequences.
%C Apart from the second diagonal (which gives the toothpick sequence A139250), the rest of the diagonals cannot be represented with toothpick structures. - _Omar E. Pol_, Dec 14 2016
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H 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>
%F See A162958 for rules governing the generation of N-th Toothpick sequences. By way of example, (N+2), A139250. The generator is A160552, which uses the multiplier "2". Then A160552 convolved with (1, 2, 2, 2,...) = A139250 the Toothpick sequence for N=2. Similarly, we create an array for Toothpick sequences N=1, 2, 3,...etc = A163267, A139250, A162958,...; then take the antidiagonals, creating triangle A163311.
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 3, 4;
%e 1, 4, 7, 5;
%e 1, 5, 10, 11, 7;
%e 1, 6, 13, 19, 15, 10;
%e 1, 7, 16, 29, 25, 23, 13;
%e 1, 8, 19, 41, 37, 40, 35, 14;
%e 1, 9, 22, 55 51, 61, 67, 43, 16;
%e 1, 10, 25, 71, 67, 86, 109, 94, 47, 19;
%e 1, 11, 28, 89, 85, 115, 161, 173, 100, 55, 22;
%e 1, 12, 31, 109, 105, 148, 223, 286, 181, 115, 67, 25;
%e 1, 13, 34, 131, 127, 185, 295, 439, 296, 205, 142, 79, 30;
%e 1, 14, 37, 155, 151, 226, 377, 638, 451, 331, 253, 175, 95, 36;
%e ...
%Y Row sums = A163312: (1, 3, 8, 17, 34, 64,...).
%Y Right border = A163267, toothpick sequence for N=1.
%Y Next diagonal going to the left = A139250, toothpick sequence for N=2.
%Y Then 1, 4, 10, 19,... = A162958, toothpick sequence for N=3.
%Y Cf. A160552, A162958, A163311, A163312.
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Jul 24 2009
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