A159912 Partial sums of A159913(k) = 2^bitcount(2k+1)-1 = A038573(2k+1), bitcount=A000120.

0, 1, 4, 7, 14, 17, 24, 31, 46, 49, 56, 63, 78, 85, 100, 115, 146, 149, 156, 163, 178, 185, 200, 215, 246, 253, 268, 283, 314, 329, 360, 391, 454, 457, 464, 471, 486, 493, 508, 523, 554, 561, 576, 591, 622, 637, 668, 699, 762, 769, 784, 799, 830, 845, 876, 907

Offset: 0

Author

M. F. Hasler, May 03 2009

Keywords

• nonn

Comments

More precisely, a(n)=sum(iA159913(i)), since we want the sequence to start with a(0)=0 and not with A159913(0)=1.

a(n) is also the total number of ON cells after n generations in the outward corner version of the Ulam-Warburton cellular automaton of A147562, and a(n) is also the total number of Y-toothpicks after n generations in the outward corner version of the Y-toothpick structure of A160120. - David Applegate and Omar E. Pol, Jan 24 2016

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• David Applegate, The movie version
• 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, pp. 6, 30.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A000120, A038573, A147562, A159913, A160120, A160720, A266532, A267700.

Programs

Mathematica

Accumulate@ Table[2^(DigitCount[n, 2][[1]] + 1) - 1, {n, 0, 54}] (* Michael De Vlieger, Jan 25 2016 *)

PARI

A159912(n)=sum(i=0,n-1,1<
				

Formula

a(n) = sum( i=0...n-1, A159913(i)) = sum(i=0..n-1, 2^A000120(i))*2-n

a(n) = n + (A160720(n) - 1)/2 = n + 2*(A266532(n) - 1)/3 = n + 2*A267700(n-1), n >= 1. - Omar E. Pol, Jan 25 2016