A129711 - OEIS

%I #3 Mar 30 2012 17:36:14

%S 1,2,2,1,3,2,5,2,1,8,3,2,13,5,2,1,21,8,3,2,34,13,5,2,1,55,21,8,3,2,89,

%T 34,13,5,2,1,144,55,21,8,3,2,233,89,34,13,5,2,1,377,144,55,21,8,3,2,

%U 610,233,89,34,13,5,2,1,987,377,144,55,21,8,3,2,1597,610,233,89,34,13,5,2,1

%N Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and starting with exactly k 01's (0<=k<=floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

%C Row n has 1+floor(n/2) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=F(n+1)-1=A000071(n+1).

%F T(n,k)=F(n-2k+1) if 2k+1<n, where F(j) are the Fibonacci numbers (F(0)=0, F(1)=1); T(2k+1,k)=2; T(n,k)=0 if 2k>n. G.f.=G(t,z)=(1+z)(1-z^2)/[(1-z-z^2)(1-tz^2)].

%e T(7,2)=3 because we have 0101110, 0101111 and 0101101.

%e Triangle starts:

%e 1;

%e 2;

%e 2,1;

%e 3,2;

%e 5,2,1;

%e 8,3,2;

%e 13,5,2,1;

%p with(combinat): T:=proc(n,k) if n=2*k+1 then 2 elif n<2*k then 0 else fibonacci(n-2*k+1) fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

%Y Cf. A000045, A000071.

%K nonn,tabf

%O 0,2

%A _Emeric Deutsch_, May 12 2007