%I #44 Mar 06 2023 18:07:12

%S 0,1,11,21,121,131,231,331,1331,1341,1441,1541,2541,2641,3641,4641,

%T 14641,14651,14751,14851,15851,15951,16951,17951,27951,28051,29051,

%U 30051,40051,41051,51051,61051,161051,161061,161161,161261,162261,162361,163361,164361

%N 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 From _Gary W. Adamson_, Aug 30 2016: (Start)

%C Let M =

%C 1, 0, 0, 0, 0, ...

%C 11, 0, 0, 0, 0, ...

%C 10, 1, 0, 0, 0, ...

%C 0, 11, 0, 0, 0, ...

%C 0, 10, 1, 0, 0, ...

%C 0, 0, 11, 0, 0, ...

%C 0, 0, 10, 1, 0, ...

%C ...

%C Then lim_{k->infinity} M^k converges to a single nonzero column giving the sequence.

%C The sequence divided by its aerated variant is (1, 11, 10, 0, 0, 0, ...). (End)

%H 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 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 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 a(n) = Sum_{k=0..n-1} 10^wt(k), where wt = A000120. - _Mike Warburton_, Mar 14 2019

%F a(n) = Sum_{k=0..floor(log_2(n))} 10^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 11*a(n/2) else 10*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..42);

%t 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 Cf. A000120, A006046, A077465, A084230.

%Y Cf. A130665, A116520, A130667, A116522, A161342, A116526, A360189.

%K nonn

%O 0,3

%A _Roger L. Bagula_, Mar 15 2006

%E Edited by _N. J. A. Sloane_, Apr 16 2005