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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 23 01:37 EST 2021. Contains 340384 sequences. (Running on oeis4.)