A129709 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 011 subwords (0<=k<=floor(n/3)). A Fibonacci binary word is a binary word having no 00 subword.

1, 2, 3, 4, 1, 5, 3, 6, 7, 7, 13, 1, 8, 22, 4, 9, 34, 12, 10, 50, 28, 1, 11, 70, 58, 5, 12, 95, 108, 18, 13, 125, 188, 50, 1, 14, 161, 308, 121, 6, 15, 203, 483, 261, 25, 16, 252, 728, 520, 80, 1, 17, 308, 1064, 968, 220, 7, 18, 372, 1512, 1710, 536, 33, 19, 444, 2100

Offset

0,2

Comments

Also number of Fibonacci binary words of length n and having k 110 subwords. Row n has 1+floor(n/3) terms. Row sums are the Fibonacci numbers (A000045). T(n,0)=n+1. Sum(k*T(n,k), k>=0)=A023610(n-3).

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

T(7,2)=4 because we have 1011011,0111011,0110110 and 0110111.
Triangle starts:
1;
2;
3;
4,1;
5,3;
6,7;
7,13,1;
8,22,4;
9,34,12;
10,50,28,1;

Maple

G:=(1+z)/(1-z-z^2+z^3-t*z^3): Gser:=simplify(series(G, z=0, 23)): for n from 0 to 20 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 20 do seq(coeff(P[n], t, j), j=0..floor(n/3)) od; # yields sequence in triangular form

Crossrefs

Cf. A000045, A023610.

Sequence in context: A071437 A243713 A332266 * A253146 A253028 A133108

Adjacent sequences: A129706 A129707 A129708 * A129710 A129711 A129712

Keyword

nonn,tabf

Author

Emeric Deutsch, May 12 2007

Status

approved