A162958 Equals A162956 convolved with (1, 3, 3, 3, ...).

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

Offset: 1

Author

Gary W. Adamson, Jul 18 2009

Keywords

• nonn

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

Crossrefs

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

Third diagonal of A163311.

Programs

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 *)

Extensions

Clarified definition by Omar E. Pol, Feb 06 2017