A105321 Convolution of binomial(1,n) and Gould's sequence A001316.

1, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 34, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 36, 12, 16, 24, 24, 24, 32, 48, 40, 24, 32, 48, 48, 48, 64, 96, 66, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16

Offset

0,2

Comments

A universal function related to the spherical growth of repeated truncations of maps.

a(n) = (number of ones in row n of triangle A249133) = (number of odd terms in row n of triangle A249095) = A000120(A249184(n)). - Reinhard Zumkeller, Nov 14 2014

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

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.]

T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Preprint series, Univ. of Ljubljana, Vol. 38 (2000), 696.

T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS - Omar E. Pol, May 29 2010

Formula

G.f. (1+x)*Product{k>=0, 1+2x^(2^k)};

a(n) = Sum_{k=0..n, binomial(1, n-k)*Sum_{j=0..k, binomial(k, j) mod 2}}.

a(n) = 2*A048460(n) for n>=2. - Omar E. Pol, Jan 02 2011

a((2*n-1)*2^p) = (2^p+2)*A001316(n-1), p >= 0 and n >= 1, with a(0) = 1. - Johannes W. Meijer, Jan 28 2013

a(n) = A001316(n) + A001316(n-1) for n > 0. - Reinhard Zumkeller, Nov 14 2014

Example

Contribution from Omar E. Pol, May 29 2010: (Start)
If written as a triangle:
1;
3;
4;
6,6;
6,8,12,10;
6,8,12,12,12,16,24,18;
6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,34;
6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,36,12,16,24,24,24,32,48,40,24,32,48,48,48,64,96,66;
(End)

Maple

nmax := 74: A001316 := n -> if n <= -1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(log[2](nmax)) do for n from 1 to nmax/(p+2)+1 do a((2*n-1)*2^p) := (2^p+2)  * A001316(n-1) od: od: a(0) :=1: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013

Mathematica

f[n_] := Sum[Binomial[1, n - k]Mod[Binomial[k, j], 2], {k, 0, n}, {j, 0, k}]; Array[f, 75, 0] (* Robert G. Wilson v, Jun 28 2010 *)

Prog

(Haskell)
a105321 n = if n == 0 then 1 else a001316 n + a001316 (n - 1)
-- Reinhard Zumkeller, Nov 14 2014
(PARI) a(n) = sum(k=0, n, binomial(1, n-k)*sum(j=0, k, binomial(k, j) % 2)); \\ Michel Marcus, Apr 29 2018

Crossrefs

Cf. A139250, A139251, A173522, A048460.

Cf. A001316, A000120, A249184, A249095, A249133.

Sequence in context: A298808 A033957 A031131 * A217032 A300305 A160095

Adjacent sequences: A105318 A105319 A105320 * A105322 A105323 A105324

Keyword

easy,nonn

Author

Paul Barry, Apr 01 2005

Status

approved