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%I A116525 #44 Mar 06 2023 18:07:12
%S A116525 0,1,11,21,121,131,231,331,1331,1341,1441,1541,2541,2641,3641,4641,
%T A116525 14641,14651,14751,14851,15851,15951,16951,17951,27951,28051,29051,
%U A116525 30051,40051,41051,51051,61051,161051,161061,161161,161261,162261,162361,163361,164361
%N A116525 a(0)=1, a(1)=1, a(n) = 11*a(n/2) for even n, and a(n) = 10*a((n-1)/2) + a((n+1)/2) for odd n >= 3.
%C A116525 From _Gary W. Adamson_, Aug 30 2016: (Start)
%C A116525 Let M =
%C A116525     1,  0,  0, 0, 0, ...
%C A116525    11,  0,  0, 0, 0, ...
%C A116525    10,  1,  0, 0, 0, ...
%C A116525     0, 11,  0, 0, 0, ...
%C A116525     0, 10,  1, 0, 0, ...
%C A116525     0,  0, 11, 0, 0, ...
%C A116525     0,  0, 10, 1, 0, ...
%C A116525   ...
%C A116525 Then lim_{k->infinity} M^k converges to a single nonzero column giving the sequence.
%C A116525 The sequence divided by its aerated variant is (1, 11, 10, 0, 0, 0, ...). (End)
%H A116525 Alois P. Heinz, <a href="/A116525/b116525.txt">Table of n, a(n) for n = 0..16383</a> (first 2501 terms from G. C. Greubel)
%H A116525 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 A116525 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 A116525 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Stolarsky-HarborthConstant.html">Stolarsky-Harborth Constant</a>
%F A116525 Let r(x) = (1 + 11x + 10x^2).  The sequence is r(x) * r(x^2) * r(x^4) * r(x^8) * ... - _Gary W. Adamson_, Aug 30 2016
%F A116525 a(n) = Sum_{k=0..n-1} 10^wt(k), where wt = A000120. - _Mike Warburton_, Mar 14 2019
%F A116525 a(n) = Sum_{k=0..floor(log_2(n))} 10^k*A360189(n-1,k). - _Alois P. Heinz_, Mar 06 2023
%p A116525 a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 11*a(n/2) else 10*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..42);
%t A116525 b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 11*b[n/2]; b[n_?OddQ] := b[n] = 10*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]
%Y A116525 Cf. A000120, A006046, A077465, A084230.
%Y A116525 Cf. A130665, A116520, A130667, A116522, A161342, A116526, A360189.
%K A116525 nonn
%O A116525 0,3
%A A116525 _Roger L. Bagula_, Mar 15 2006
%E A116525 Edited by _N. J. A. Sloane_, Apr 16 2005

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