A238438 Expansion of 1/G(0) where G(k) = 1 - q/(1 - q - q^3 / G(k+1) ).

1, 1, 2, 4, 9, 21, 50, 121, 297, 738, 1853, 4694, 11982, 30790, 79586, 206786, 539784, 1414905, 3722776, 9828501, 26028969, 69129150, 184076913, 491340306, 1314412198, 3523519135, 9463563168, 25462981484, 68626114915, 185246103584, 500779373140, 1355636896041, 3674558399538, 9972405246294, 27095580261125

Offset: 0

Author

Joerg Arndt, Feb 27 2014

Keywords

• nonn

Comments

What does this sequence count?

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A086581: 1/G(0) where G(k) = 1 - q/(1 - q - q^2 / G(k+1) ).

Cf. A119370: 1/G(0) where G(k) = 1 - q/(1 - (q + q^2) / G(k+1) ).

Programs

Mathematica

CoefficientList[Series[2*(1-x)/(1 - 2*x + x^3 + Sqrt[1 - 4*x + 4*x^2 - 2*x^3 + x^6]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 01 2014 *)

PARI

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1,  1 - q/(1 - q - q^3 / G(k+1) ) );
Vec( 1/G(0) )

Formula

From Vaclav Kotesovec, Mar 01 2014: (Start)

G.f.: 2*(1-x)/(1 - 2*x + x^3 + sqrt(1 - 4*x + 4*x^2 - 2*x^3 + x^6)).

D-finite with Recurrence: (n+3)*a(n) = 2*(2*n+3)*a(n-1) - 4*n*a(n-2) + (2*n-3)*a(n-3) - (n-6)*a(n-6).

a(n) ~ (6*r^2+14*r+17) * sqrt(7*r-2) / (2 * sqrt(Pi) * n^(3/2) * r^(n-1/2)), where r = 1/3*(-2 - 2*(2/(47 + 3*sqrt(249)))^(1/3) + (1/2*(47 + 3*sqrt(249)))^(1/3)) = 0.3532099641993244294831... is the root of the equation r^3 + 2*r^2 + 2*r = 1.

(End)

G.f. A(q) satisfies 0 = -q^3*A(q)^2 + (q^3 - 2*q + 1)*A(q) + (q - 1).