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.

0, 1, 11, 21, 121, 131, 231, 331, 1331, 1341, 1441, 1541, 2541, 2641, 3641, 4641, 14641, 14651, 14751, 14851, 15851, 15951, 16951, 17951, 27951, 28051, 29051, 30051, 40051, 41051, 51051, 61051, 161051, 161061, 161161, 161261, 162261, 162361, 163361, 164361

Offset

0,3

Comments

From Gary W. Adamson, Aug 30 2016: (Start)

Let M =

1, 0, 0, 0, 0, ...

11, 0, 0, 0, 0, ...

10, 1, 0, 0, 0, ...

0, 11, 0, 0, 0, ...

0, 10, 1, 0, 0, ...

0, 0, 11, 0, 0, ...

0, 0, 10, 1, 0, ...

...

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)

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

Formula

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

a(n) = Sum_{k=0..n-1} 10^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

a(n) = Sum_{k=0..floor(log_2(n))} 10^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Maple

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);

Mathematica

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}]

Crossrefs

Cf. A000120, A006046, A077465, A084230.

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

Sequence in context: A096104 A126299 A166707 * A094623 A321509 A034922

Adjacent sequences: A116522 A116523 A116524 * A116526 A116527 A116528

Keyword

nonn

Author

Roger L. Bagula, Mar 15 2006

Extensions

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

Status

approved