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.

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

0,2

Comments

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).

Links

Table of n, a(n) for n=0..75.

Formula

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].

Example

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;

Maple

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

Crossrefs

Cf. A000045, A129719, A129722.

Sequence in context: A276468 A002126 A350815 * A268193 A238606 A325612

Adjacent sequences: A129718 A129719 A129720 * A129722 A129723 A129724

Keyword

nonn,tabf

Author

Emeric Deutsch, May 13 2007

Status

approved