A129719 -id:A129719 - cp's OEIS frontend

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

Emeric Deutsch, May 13 2007

nonn

			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 ').

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).

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

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);

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 *)

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

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

A129720 Number of 0's in odd position in all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

0, 1, 1, 4, 5, 14, 19, 46, 65, 145, 210, 444, 654, 1331, 1985, 3926, 5911, 11434, 17345, 32960, 50305, 94211, 144516, 267384, 411900, 754309, 1166209, 2116936, 3283145, 5914310, 9197455, 16458034, 25655489, 45638101, 71293590, 126159156

Offset: 0

Emeric Deutsch, May 13 2007

nonn

			a(4)=5 because in 1110, 1111, 110'1, 1010, 1011, 0'110, 0'111 and 0'10'1 one has altogether five 0's in odd position (marked by ').

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

Cf. A030267, A001870, A129719, A129722.

g:=z*(1-z^2)/(1-z-z^2)^2/(1+z-z^2): gser:=series(g,z=0,43): seq(coeff(gser,z,n),n=0..40);

G.f.: z(1-z^2)/((1-z-z^2)^2*(1+z-z^2)).
a(2n) = a(2n-1) + a(2n-2) (n >= 1).
a(2n-1) = A030267(n).
a(2n) = A129722(2n) = A001870(n-1).
a(n) = Sum_{k=0..ceiling(n/2)} k*A129719(n,k).

A129721 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 0's in even positions (0<=k<=floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 2, 1, 4, 1, 4, 3, 1, 8, 4, 1, 8, 8, 4, 1, 16, 12, 5, 1, 16, 20, 13, 5, 1, 32, 32, 18, 6, 1, 32, 48, 38, 19, 6, 1, 64, 80, 56, 25, 7, 1, 64, 112, 104, 63, 26, 7, 1, 128, 192, 160, 88, 33, 8, 1, 128, 256, 272, 192, 96, 34, 8, 1, 256, 448, 432, 280, 129, 42, 9, 1, 256, 576, 688, 552

Offset: 0

Emeric Deutsch, May 13 2007

Row n has 1+floor(n/2) terms. Row sums are the Fibonacci numbers (A000045). T(2n+1,k)=T(2n,k)+T(2n-1,k) (n>=1). T(2n,k)=A129719(2n,k). Sum(k*T(n,k), 0<=k<=floor(n/2))=A129722(n).

			T(6,2)=4 because we have 111010, 101110, 101011 and 011010.
Triangle starts:
1;
2;
2,1;
4,1;
4,3,1;
8,4,1;
8,8,4,1;

Cf. A000045, A129719, A129722.

G:=(1+2*z-t*z^3)/(1-2*z^2-t*z^2+t*z^4): Gser:=simplify(series(G,z=0,21)): for n from 0 to 18 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

G.f.=G(t,z)=(1+2z-tz^3)/[1-(2+t)z^2+tz^4]. The trivariate generating function H(t,s,z), where t marks number of 0's in odd position and s marks number of 0's in even position, is given by H(t,s,z)=[1+(1+t)z-tsz^3]/[1-(1+t+s)z^2+tsz^4].

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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.

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A129720 Number of 0's in odd position in all Fibonacci binary words of length n. A Fibonacci binary word is a binary word having no 00 subword.

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A129721 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 0's in even positions (0<=k<=floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

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