A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

1, 1, 2, 5, 13, 35, 95, 260, 713, 1959, 5386, 14815, 40759, 112151, 308609, 849240, 2337009, 6431246, 17698332, 48704714, 134032593, 368850417, 1015056867, 2793383746, 7687248186, 21154913043, 58217239536, 160210872557, 440892153268, 1213312738702, 3338974845151, 9188688696438

Offset: 0

Author

Joerg Arndt, Jul 06 2013

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).

Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).

Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).

Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).

Cf. A227310 (g.f.: 1/G(0), where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).

Programs

Mathematica

nmax = 40; CoefficientList[Series[1/(1 - x/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 2] - Floor[Range[nmax + 2]/2])]]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

PARI

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ) );
gf = 1 /(1- q/G(0));
Vec(gf)

Formula

G.f.: 1/(1-q/ (1-q/(1-q/ (1-q^2/(1-q^2/ (1-q^3/(1-q^3/ (1-q^4/(1-q^4/ (1-q^5/(1-q^5/ (1-...))))))))))) ).

G.f. A(x) = 1/(1 - B(x)) where B(x) is the g.f. of A006958.

a(n) ~ c * d^n, where d = 2.751949072495748078279227332764623096815571855905843246297955690122791154... and c = 0.215973947378529032758849789768859077066690378163074586384819930605436492... - Vaclav Kotesovec, Sep 05 2017