cp's OEIS Frontend

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.

%I A129712 #2 Mar 30 2012 17:36:14
%S A129712 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,
%T A129712 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,
%U A129712 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
%N 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.
%C A129712 Row n has 1+floor(n/2) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=A052952(n-2) (n>=2).
%F A129712 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(n-2k) for 1<=k<(n-1)/2. G.f.=G(t,z)=(1+z-z^2-t*z^3)/[(1-z-z^2)(1-t*z^2)].
%e A129712 T(7,2)=2 because we have 1010110 and 1010111.
%e A129712 Triangle starts:
%e A129712 1;
%e A129712 2;
%e A129712 2,1;
%e A129712 4,1;
%e A129712 6,1,1;
%e A129712 10,2,1;
%e A129712 16,3,1,1;
%e A129712 26,5,2,1;
%p A129712 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(n-2*k) else 0 fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od;
%Y A129712 Cf. A000045, A052952.
%K A129712 nonn,tabf
%O A129712 0,2
%A A129712 _Emeric Deutsch_, May 12 2007