

A129712


Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 10's (0<=k<=floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.


0



1, 2, 2, 1, 4, 1, 6, 1, 1, 10, 2, 1, 16, 3, 1, 1, 26, 5, 2, 1, 42, 8, 3, 1, 1, 68, 13, 5, 2, 1, 110, 21, 8, 3, 1, 1, 178, 34, 13, 5, 2, 1, 288, 55, 21, 8, 3, 1, 1, 466, 89, 34, 13, 5, 2, 1, 754, 144, 55, 21, 8, 3, 1, 1, 1220, 233, 89, 34, 13, 5, 2, 1, 1974, 377, 144, 55, 21, 8, 3, 1, 1
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OFFSET

0,2


COMMENTS

Row n has 1+floor(n/2) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=A052952(n2) (n>=2).


LINKS

Table of n, a(n) for n=0..80.


FORMULA

T(0,0)=1, T(n,0)=2F(n) for n>=1, T(2k,k)=T(2k+1,k)=1 for k>=1, T(n,k)=F(n2k) for 1<=k<(n1)/2. G.f.=G(t,z)=(1+zz^2t*z^3)/[(1zz^2)(1t*z^2)].


EXAMPLE

T(7,2)=2 because we have 1010110 and 1010111.
Triangle starts:
1;
2;
2,1;
4,1;
6,1,1;
10,2,1;
16,3,1,1;
26,5,2,1;


MAPLE

with(combinat): T:=proc(n, k) if k=0 and n=0 then 1 elif k=0 then 2*fibonacci(n) elif n=2*k or n=2*k+1 then 1 elif n>2*k+1 then fibonacci(n2*k) else 0 fi end: for n from 0 to 18 do seq(T(n, k), k=0..floor(n/2)) od;


CROSSREFS

Cf. A000045, A052952.
Sequence in context: A116633 A263232 A134666 * A051720 A022693 A174498
Adjacent sequences: A129709 A129710 A129711 * A129713 A129714 A129715


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, May 12 2007


STATUS

approved



