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

0,2

Comments

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

Links

Table of n, a(n) for n=0..64.

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).

Example

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;
  ...

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 *)

Crossrefs

Cf. A000045, A001629, A130137.

Sequence in context: A321128 A130138 A171855 * A197124 A032759 A142711

Adjacent sequences: A130133 A130134 A130135 * A130137 A130138 A130139

Keyword

nonn,tabf

Author

Emeric Deutsch, May 13 2007

Status

approved