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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,....

%I A116524 #29 Feb 16 2025 08:33:00
%S A116524 0,1,13,25,169,181,325,469,2197,2209,2353,2497,4225,4369,6097,7825,
%T A116524 28561,28573,28717,28861,30589,30733,32461,34189,54925,55069,56797,
%U A116524 58525,79261,80989,101725,122461,371293,371305,371449,371593,373321
%N 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,....
%C A116524 A 13-divide version of A084230.
%C A116524 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.
%H A116524 G. C. Greubel, <a href="/A116524/b116524.txt">Table of n, a(n) for n = 0..2500</a>
%H A116524 H. Harborth, <a href="http://dx.doi.org/10.1090/S0002-9939-1977-0429714-1">Number of Odd Binomial Coefficients</a>, Proc. Amer. Math. Soc. 62, 19-22, 1977.
%H A116524 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
%H A116524 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Stolarsky-HarborthConstant.html">Stolarsky-Harborth Constant</a>
%F A116524 a(n) = Sum_{k=0..n-1} 12^wt(k), where wt = A000120. - _Mike Warburton_, Mar 14 2019
%p A116524 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);
%t A116524 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}]
%Y A116524 Cf. A000120, A006046, A077465, A084230.
%K A116524 nonn,look
%O A116524 0,3
%A A116524 _Roger L. Bagula_, Mar 15 2006
%E A116524 Edited by _N. J. A. Sloane_, Apr 16 2005