A116523 a(0)=1, a(1)=1, a(n) = 17*a(n/2) for n=2,4,6,..., a(n) = 16*a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

0, 1, 17, 33, 289, 305, 561, 817, 4913, 4929, 5185, 5441, 9537, 9793, 13889, 17985, 83521, 83537, 83793, 84049, 88145, 88401, 92497, 96593, 162129, 162385, 166481, 170577, 236113, 240209, 305745, 371281, 1419857, 1419873, 1420129, 1420385

Offset: 0

Author

Roger L. Bagula, Mar 15 2006

Keywords

• nonn

Comments

A 17-divide version of A084230.

The Harborth : f(2^k)=3^k suggests that a family of sequences of the form: f(2^k)=Prime[n]^k There does indeed seem to be an infinite family of such functions.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
• 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. 27, 33.
• Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Crossrefs

Cf. A000120, A006046, A077465, A084230.

Programs

Maple

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 17*a(n/2) else 16*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..38);

Mathematica

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 17*b[n/2]; b[n_?OddQ] := b[n] = 16*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

Formula

a(n) = Sum_{k=0..n-1} 16^wt(k), where wt = A000120. - Mike Warburton, Mar 22 2019

Extensions

Edited by N. J. A. Sloane, Apr 16 2005