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A162958 Equals A162956 convolved with (1, 3, 3, 3, ...). 5
1, 4, 10, 19, 25, 40, 67, 94, 100, 115, 142, 175, 208, 280, 388, 469, 475, 490, 517, 550, 583, 655, 763, 850, 883, 955, 1069, 1201, 1372, 1696, 2101, 2344, 2350, 2365, 2392, 2425, 2458, 2530, 2638, 2725, 2758, 2830, 2944, 3076, 3247, 3571, 3976, 4225, 4258 (list; graph; refs; listen; history; text; internal format)
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, ...).

Examples: A160552 * (1, 2, 0, 0, 0,...) = a B-type sequence A151548.

A160552 * (1, 2, 2, 2, 2,...) = the toothpick sequence A139250 for N=2.

A162956 is analogous to A160552 but replaces "2" with the multiplier "3".

Row terms of A162956 tend to A162957 = (1, 3, 0, 0, 0, ...) * A162956.

Toothpick sequence for N = 3 = A162958 = A162956 * (1, 3, 3, 3, ...).

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-Fran├žois Alcover, Jun 11 2018, after Alois P. Heinz *)

CROSSREFS

Cf. A139250, A152548, A160552, A162956, A163267.

Third diagonal of A163311.

Sequence in context: A318070 A073262 A145731 * A029714 A062198 A050858

Adjacent sequences:  A162955 A162956 A162957 * A162959 A162960 A162961

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jul 18 2009

EXTENSIONS

Clarified definition by Omar E. Pol, Feb 06 2017

STATUS

approved

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Last modified January 18 21:54 EST 2019. Contains 319282 sequences. (Running on oeis4.)