A130138 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 1011's (n>=0, 0<=k<=floor((n-1)/3)). A Fibonacci binary word is a binary word having no 00 subword.

1, 2, 3, 5, 7, 1, 9, 4, 11, 10, 13, 20, 1, 15, 35, 5, 17, 56, 16, 19, 84, 40, 1, 21, 120, 86, 6, 23, 165, 166, 23, 25, 220, 296, 68, 1, 27, 286, 496, 171, 7, 29, 364, 791, 382, 31, 31, 455, 1211, 781, 105, 1, 33, 560, 1792, 1488, 300, 8, 35, 680, 2576, 2678, 756, 40, 37

Offset

0,2

Comments

Row n has 1+floor((n-1)/3) terms. Row sums are the Fibonacci numbers (A000045). T(n,0)=A004280(n+1). Sum(k*T(n,k), k>=0)=A004798(n-3) (n>=4).

Links

Table of n, a(n) for n=0..64.

Formula

G.f.=G(t,z)=(1+z)(1+z^3-tz^3)/[1-z-z^2+z^3-tz^3].

Example

T(7,2)=1 because we have 1011011.
Triangle starts:
1;
2;
3;
5;
7,1;
9,4;
11,10;
13,20,1;
15,35,5;

Maple

G:=(1+z)*(1+z^3-t*z^3)/(1-z-z^2+z^3-t*z^3): Gser:=simplify(series(G, z=0, 24)): for n from 0 to 21 do P[n]:=sort(coeff(Gser, z, n)) od: 1; for n from 1 to 21 do seq(coeff(P[n], t, j), j=0..floor((n-1)/3)) od; # yields sequence in triangular form

Crossrefs

Cf. A000045, A004280, A004798.

Sequence in context: A178743 A126052 A321128 * A171855 A130136 A197124

Adjacent sequences: A130135 A130136 A130137 * A130139 A130140 A130141

Keyword

nonn,tabf

Author

Emeric Deutsch, May 13 2007

Status

approved