A116524 a(0)=1, a(1)=1, a(n) = 13*a(n/2) for n=2,4,6,..., a(n) = 12*a((n-1)/2) + a((n+1)/2) for n=3,5,7,....

0, 1, 13, 25, 169, 181, 325, 469, 2197, 2209, 2353, 2497, 4225, 4369, 6097, 7825, 28561, 28573, 28717, 28861, 30589, 30733, 32461, 34189, 54925, 55069, 56797, 58525, 79261, 80989, 101725, 122461, 371293, 371305, 371449, 371593, 373321

Offset: 0

Author

Roger L. Bagula, Mar 15 2006

Keywords

• nonn
• look

Comments

A 13-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 13*a(n/2) else 12*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..40);

Mathematica

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 13*b[n/2]; b[n_?OddQ] := b[n] = 12*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

Formula

a(n) = Sum_{k=0..n-1} 12^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

Extensions

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