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%I #36 Mar 06 2023 18:03:56
%S 0,1,9,17,81,89,153,217,729,737,801,865,1377,1441,1953,2465,6561,6569,
%T 6633,6697,7209,7273,7785,8297,12393,12457,12969,13481,17577,18089,
%U 22185,26281,59049,59057,59121,59185,59697,59761,60273,60785,64881,64945,65457,65969
%N a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3.
%C A 9-divide version of A084230.
%C The interest this one has is in the prime form of even odd 2^n+1, 2^n.
%C From _Gary W. Adamson_, Aug 30 2016: (Start)
%C Let M =
%C 1, 0, 0, 0, 0, ...
%C 9, 0, 0, 0, 0, ...
%C 8, 1, 0, 0, 0, ...
%C 0, 9, 0, 0, 0, ...
%C 0, 8, 1, 0, 0, ...
%C 0, 0, 9, 0, 0, ...
%C 0, 0, 8, 1, 0, ...
%C ...
%C Then M^k converges to a single nonzero column giving the sequence.
%C The sequence divided by its aerated variant is (1, 9, 8, 0, 0, 0, ...). (End)
%H Alois P. Heinz, <a href="/A116526/b116526.txt">Table of n, a(n) for n = 0..16383</a> (first 2501 terms from G. C. Greubel)
%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Stolarsky-HarborthConstant.html">Stolarsky-Harborth Constant</a>
%F a(n) = Sum_{k=0..n-1} 8^wt(k), where wt = A000120. - _Mike Warburton_, Mar 14 2019
%F a(n) = Sum_{k=0..floor(log_2(n))} 8^k*A360189(n-1,k). - _Alois P. Heinz_, Mar 06 2023
%p a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 9*a(n/2) else 8*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..45);
%t b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 9*b[n/2]; b[n_?OddQ] := b[n] = 8*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]
%Y Cf. A000120, A006046, A077465, A130665, A116520, A130667, A116522, A161342, A116525, A360189.
%K nonn
%O 0,3
%A _Roger L. Bagula_, Mar 15 2006
%E Edited by _N. J. A. Sloane_, Apr 16 2006
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