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

Author

Emeric Deutsch, May 13 2007

Keywords

• nonn

Examples

			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 ').
		

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).

Crossrefs

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

Programs

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 *)

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

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