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

Author

Paul Barry, Apr 01 2005

Keywords

• easy
• nonn

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

Examples

			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)
		

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

Crossrefs

Cf. A139250, A139251, A173522, A048460.

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

Programs

Haskell

a105321 n = if n == 0 then 1 else a001316 n + a001316 (n - 1)
-- Reinhard Zumkeller, Nov 14 2014

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 *)

PARI

a(n) = sum(k=0, n, binomial(1, n-k)*sum(j=0, k, binomial(k, j) % 2)); \\ Michel Marcus, Apr 29 2018

Python

def A105321(n): return (1<Chai Wah Wu, Jul 30 2025

Python

# (fast way for big vectors)
import numpy # (version >= 2.0.0)
n_up_to = 2**22
A000079 = 1 << numpy.arange(n_up_to.bit_length())
A001316 = A000079[(numpy.bitwise_count(numpy.arange(n_up_to)))]
A105321 = A001316
A105321[1:] += A001316[0:-1]
print(A105321[0:100]) # Karl-Heinz Hofmann, Aug 01 2025

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