A151917 a(0)=0, a(1)=1; for n>=2, a(n) = (2/3)*(Sum_{i=1..n-1} 3^wt(i)) + 1, where wt() = A000120().

0, 1, 3, 5, 11, 13, 19, 25, 43, 45, 51, 57, 75, 81, 99, 117, 171, 173, 179, 185, 203, 209, 227, 245, 299, 305, 323, 341, 395, 413, 467, 521, 683, 685, 691, 697, 715, 721, 739, 757, 811, 817, 835, 853, 907, 925, 979, 1033, 1195, 1201, 1219

Offset: 0

Author

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Keywords

• nonn
• easy

Comments

Also, total number of "ON" cells at n-th stage in two of the four wedges of the "Ulam-Warburton" two-dimensional cellular automaton of A147562, but including the central ON cell. It appears that this is very close to A139250, the toothpick sequence. - Omar E. Pol, Feb 22 2015

Examples

			n=3: (2/3)*(3^1+3^1+3^2+3^1) + 1 = (2/3)*18 + 1 = 13.
		

Links

• Michel Marcus, Table of n, a(n) for n = 0..10000
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A000120, A007583, A139250, A147562, A151914, A151920.

Programs

Mathematica

Array[(2/3) Sum[3^(Total@ IntegerDigits[i, 2]), {i, # - 1}] + 1 &, 50] (* Michael De Vlieger, Nov 01 2022 *)

PARI

a(n) = if (n<2, n, 1 + 2*sum(i=1,n-1, 3^hammingweight(i))/3); \\ Michel Marcus, Feb 22 2015

Formula

a(n) = A151914(n)/4.

a(n) = A079315(2n)/4.

For n>=2, a(n) = 2*A151920(n-2) + 1.

For n>=1, a(n) = (1 + A147562(n))/2. - Omar E. Pol, Mar 13 2011

a(2^k) = A007583(k), if k >= 0. - Omar E. Pol, Feb 22 2015