OFFSET

1,2

COMMENTS

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".

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.

Then generalized toothpick sequences (C) = A convolved with (1, N, N, N, ...).

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, ...).

Is there an illustration of this sequence using toothpicks? - Omar E. Pol, Dec 13 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..16384

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

MAPLE

b:= proc(n) option remember; `if`(n<2, n,

(j-> 3*b(j)+b(j+1))(n-2^ilog2(n)))

end:

a:= proc(n) option remember;

`if`(n=0, 0, a(n-1)+2*b(n-1)+b(n))

end:

seq(a(n), n=1..100); # Alois P. Heinz, Jan 28 2017

MATHEMATICA

b[n_] := b[n] = If[n<2, n, Function[j, 3*b[j]+b[j+1]][n-2^Floor[Log[2, n]] ]];

a[n_] := a[n] = If[n == 0, 0, a[n-1] + 2*b[n-1] + b[n]];

Array[a, 100] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

CROSSREFS

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jul 18 2009

EXTENSIONS

Clarified definition by Omar E. Pol, Feb 06 2017

STATUS

approved