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%I #28 Feb 24 2021 02:48:18
%S 1,4,10,19,25,40,67,94,100,115,142,175,208,280,388,469,475,490,517,
%T 550,583,655,763,850,883,955,1069,1201,1372,1696,2101,2344,2350,2365,
%U 2392,2425,2458,2530,2638,2725,2758,2830,2944,3076,3247,3571,3976,4225,4258
%N Equals A162956 convolved with (1, 3, 3, 3, ...).
%C Can be considered a toothpick sequence for N=3, following rules analogous to those in A160552 (= special case of "A"), A151548 = special case "B", and the toothpick sequence A139250 (N=2) = special case "C".
%C To obtain the infinite set of toothpick sequences, (N = 2, 3, 4, ...), replace the multiplier "2" in A160552 with any N, getting a triangle with 2^n terms. Convolve this A sequence with (1, N, 0, 0, 0, ...) = B such that row terms of A triangles converge to B.
%C Then generalized toothpick sequences (C) = A convolved with (1, N, N, N, ...).
%C Examples: A160552 * (1, 2, 0, 0, 0,...) = a B-type sequence A151548.
%C A160552 * (1, 2, 2, 2, 2,...) = the toothpick sequence A139250 for N=2.
%C A162956 is analogous to A160552 but replaces "2" with the multiplier "3".
%C Row terms of A162956 tend to A162957 = (1, 3, 0, 0, 0, ...) * A162956.
%C Toothpick sequence for N = 3 = A162958 = A162956 * (1, 3, 3, 3, ...).
%C Row sums of "A"-type triangles = powers of (N+2); since row sums of A160552 = (1, 4, 16, 64, ...), while row sums of A162956 = (1, 5, 25, 125, ...).
%C Is there an illustration of this sequence using toothpicks? - _Omar E. Pol_, Dec 13 2016
%H Alois P. Heinz, <a href="/A162958/b162958.txt">Table of n, a(n) for n = 1..16384</a>
%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>
%p b:= proc(n) option remember; `if`(n<2, n,
%p (j-> 3*b(j)+b(j+1))(n-2^ilog2(n)))
%p end:
%p a:= proc(n) option remember;
%p `if`(n=0, 0, a(n-1)+2*b(n-1)+b(n))
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jan 28 2017
%t b[n_] := b[n] = If[n<2, n, Function[j, 3*b[j]+b[j+1]][n-2^Floor[Log[2, n]] ]];
%t a[n_] := a[n] = If[n == 0, 0, a[n-1] + 2*b[n-1] + b[n]];
%t Array[a, 100] (* _Jean-François Alcover_, Jun 11 2018, after _Alois P. Heinz_ *)
%Y Cf. A139250, A152548, A160552, A162956, A163267.
%Y Third diagonal of A163311.
%K nonn
%O 1,2
%A _Gary W. Adamson_, Jul 18 2009
%E Clarified definition by _Omar E. Pol_, Feb 06 2017
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