A171753 Expansion of g.f. 1/(1-3*x-x^2/(1-3*x-x^2/(1-3*x))).

1, 3, 10, 36, 137, 543, 2218, 9264, 39329, 168939, 731770, 3188364, 13948745, 61196775, 269007994, 1184076216, 5216618369, 22996827795, 101421591466, 447422614068, 1974197123657, 8712062181999, 38449506441994, 169702143024768, 749034931995041, 3306200447618043

Offset: 0

Author

Paul Barry, Dec 17 2009

Keywords

• easy
• nonn

Comments

3rd binomial transform of 1,0,1,0,2,0,4,0,8,0,...

Links

• Stefano Spezia, Table of n, a(n) for n = 0..1500
• Index entries for linear recurrences with constant coefficients, signature (9,-25,21).

Crossrefs

Cf. A083878.

Programs

Mathematica

LinearRecurrence[{9,-25,21},{1,3,10},26] (* Stefano Spezia, May 11 2024 *)

Formula

G.f.: (1-6x+8x^2)/(1-9x+25x^2-21x^3) = -(4*x-1)*(2*x-1)/((3*x-1)*(7*x^2-6*x+1)).

a(n) = (3-sqrt(2))^n/4 + (3+sqrt(2))^n/4 + 3^n/2.

a(n) = (3^n+A083878(n))/2. - R. J. Mathar, Oct 08 2016

E.g.f.: exp(3*x)*cosh(x/sqrt(2))^2. - Stefano Spezia, May 11 2024