A129722 Number of 0's in even position in all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

0, 0, 1, 1, 5, 6, 19, 25, 65, 90, 210, 300, 654, 954, 1985, 2939, 5911, 8850, 17345, 26195, 50305, 76500, 144516, 221016, 411900, 632916, 1166209, 1799125, 3283145, 5082270, 9197455, 14279725, 25655489, 39935214, 71293590, 111228804, 197452746, 308681550

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Moussa Benoumhani, On the Modes of the Independence Polynomial of the Centipede, Journal of Integer Sequences, Vol. 15 (2012), #12.5.1.

É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-4,1,1).

Formula

G.f.: z^2/( (1+z-z^2)*(1-z-z^2)^2 ).

a(2*n+1) = a(2*n) + a(2*n-1) (n>=1).

a(2*n+1) = A001871(n-1) (n>=1).

a(2*n) = A129720(2*n) = A001870(n-1).

a(n) = Sum_{ k=0..floor(n/2)} k*A129721(n,k).

a(n) = F(n)*(n+1)/5 + F(n+1)*(2*n - 5 + 5*(-1)^n)/20 where F = A000045. - Greg Dresden, Jan 01 2021

Example

a(4)=5 because in 1110', 1111, 1101, 10'10', 10'11, 0110', 0111 and 0101 one has altogether five 0's in even position (marked by ').

Maple

G:=z^2/(1-z-z^2)^2/(1+z-z^2): Gser:=series(G, z=0, 45): seq(coeff(Gser, z, n), n=0..42);

Mathematica

CoefficientList[Series[x^2/((1 + x - x^2)*(1 - x - x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Mar 09 2017 *)
LinearRecurrence[{1, 4, -3, -4, 1, 1}, {0, 0, 1, 1, 5, 6}, 40] (* Harvey P. Dale, Apr 02 2018 *)

Prog

(PARI) x='x+O('x^50); concat([0, 0], Vec(x^2/((1 + x - x^2)*(1 - x - x^2)^2))) \\ G. C. Greubel, Mar 09 2017

Crossrefs

Cf. A000045, A001870, A001871, A129719, A129720, A129721.

Sequence in context: A290254 A163772 A056509 * A133608 A317444 A072577

Adjacent sequences: A129719 A129720 A129721 * A129723 A129724 A129725

Keyword

nonn

Author

Emeric Deutsch, May 13 2007

Status

approved