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A130136 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 0110's (n>=0, 0<=k<=floor((n-1)/3)). A Fibonacci binary word is a binary word having no 00 subword.

%I A130136 #11 Aug 25 2021 06:59:09
%S A130136 1,2,3,5,7,1,11,2,16,5,25,8,1,37,16,2,57,26,6,85,48,10,1,130,78,23,2,
%T A130136 195,136,39,7,297,220,80,12,1,447,371,136,31,2,679,598,258,54,8,1024,
%U A130136 987,437,121,14,1,1553,1584,790,212,40,2,2345,2576,1332,432,71,9,3553
%N A130136 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 0110's (n>=0, 0<=k<=floor((n-1)/3)). A Fibonacci binary word is a binary word having no 00 subword.
%C A130136 Row n has 1+floor((n-1)/3) terms. Row sums are the Fibonacci numbers (A000045).
%F A130136 G.f.: G(t,z) = (1+z+(1-t)z^3)/(1-z-z^2+(1-t)z^3-(1-t)z^4).
%F A130136 T(n,0) = A130137(n).
%F A130136 Sum_{k>=0} k*T(n,k) = A001629(n-2) (n>=2).
%e A130136 T(8,2)=2 because we have 01101101 and 10110110.
%e A130136 Triangle starts:
%e A130136    1;
%e A130136    2;
%e A130136    3;
%e A130136    5;
%e A130136    7, 1;
%e A130136   11, 2;
%e A130136   16, 5;
%e A130136   25, 8, 1;
%e A130136   ...
%p A130136 G:=(1+z+z^3-t*z^3)/(1-z-z^2+z^3-t*z^3-z^4+t*z^4): Gser:=simplify(series(G,z=0,23)): for n from 0 to 23 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 20 do seq(coeff(P[n],t,j),j=0..floor((n-1)/3)) od; # yields sequence in triangular form
%t A130136 gf = (1 + z + (1-t) z^3)/(1 - z - z^2 + (1-t) z^3 - (1-t) z^4);
%t A130136 CoefficientList[#, t]& /@ CoefficientList[gf + O[z]^20, z] // Flatten (* _Jean-François Alcover_, Aug 25 2021 *)
%Y A130136 Cf. A000045, A001629, A130137.
%K A130136 nonn,tabf
%O A130136 0,2
%A A130136 _Emeric Deutsch_, May 13 2007