A135934 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - Fibonacci(k)*x).

1, 1, 2, 4, 9, 24, 77, 299, 1419, 8312, 60452, 547939, 6213566, 88468601, 1585646789, 35846274127, 1023893974778, 37005881297226, 1694206791508891, 98335493373334998, 7241161595237290969, 676871453643079089963, 80351261743964014059133, 12117563014768206457325416

Offset

0,3

Comments

After the first term, row sums of triangle A111669. - Emanuele Munarini, Dec 05 2017

Links

Alois P. Heinz, Table of n, a(n) for n = 0..140

Formula

G.f.: (1 - G(0) )/(1-x) where G(k) = 1 - 1/(1-Fibonacci(k)*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013

G.f.: 1/(x*(1-x)*G(0)) - 1/x where G(k) = 1 - x/(x - 1/(1 + 1/(x*Fibonacci(k)-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 13 2013

Example

A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-x)) + x^3/((1-x)*(1-x)*(1-2*x)) +
x^4/((1-x)*(1-x)*(1-2*x)(1-3*x)) + x^5/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)) + x^6/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)*(1-8*x)) +...

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      (<<0|1>, <1|1>>^m)[1, 2]*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2021

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, 1,
   MatrixPower[{{0, 1}, {1, 1}}, m][[1, 2]]*b[n-1, m]+b[n-1, m+1]];
a[n_] :=  b[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 07 2022, after Alois P. Heinz *)

Prog

(PARI) {a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-fibonacci(j)*x+x*O(x^n))), n)}

Crossrefs

Cf. A000045, A006116, A111669.

Sequence in context: A000667 A131351 A091352 * A343032 A210342 A137154

Adjacent sequences: A135931 A135932 A135933 * A135935 A135936 A135937

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2007

Status

approved