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.

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, 195, 136, 39, 7, 297, 220, 80, 12, 1, 447, 371, 136, 31, 2, 679, 598, 258, 54, 8, 1024, 987, 437, 121, 14, 1, 1553, 1584, 790, 212, 40, 2, 2345, 2576, 1332, 432, 71, 9, 3553

Offset: 0

Author

Emeric Deutsch, May 13 2007

Keywords

• nonn
• tabf

Comments

Row n has 1+floor((n-1)/3) terms. Row sums are the Fibonacci numbers (A000045).

Examples

			T(8,2)=2 because we have 01101101 and 10110110.
Triangle starts:
   1;
   2;
   3;
   5;
   7, 1;
  11, 2;
  16, 5;
  25, 8, 1;
  ...
		

Crossrefs

Cf. A000045, A001629, A130137.

Programs

Maple

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

Mathematica

gf = (1 + z + (1-t) z^3)/(1 - z - z^2 + (1-t) z^3 - (1-t) z^4);
CoefficientList[#, t]& /@ CoefficientList[gf + O[z]^20, z] // Flatten (* Jean-François Alcover, Aug 25 2021 *)

Formula

G.f.: G(t,z) = (1+z+(1-t)z^3)/(1-z-z^2+(1-t)z^3-(1-t)z^4).

T(n,0) = A130137(n).

Sum_{k>=0} k*T(n,k) = A001629(n-2) (n>=2).