A163470 a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0) = 3, a(1) = 15.

3, 15, 81, 453, 2571, 14679, 84009, 481245, 2757843, 15806559, 90600513, 519318837, 2976744027, 17062807335, 97804786329, 560621795277, 3213512139939, 18420013780911, 105584452428081, 605215440272805

Offset: 0

Author

Klaus Brockhaus, Aug 11 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A083881 without initial 1. Inverse binomial transform of A163471.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-13).

Crossrefs

Cf. A083881, A163471.

Programs

Magma

[ n le 2 select 12*n-9 else 8*Self(n-1)-13*Self(n-2): n in [1..22] ];

Mathematica

LinearRecurrence[{8, -13}, {3, 15}, 50] (* G. C. Greubel, Jul 25 2017 *)

PARI

x='x+O('x^50); Vec((3-9*x)/(1-8*x+13*x^2)) \\ G. C. Greubel, Jul 25 2017

Formula

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

G.f.: (3-9*x)/(1-8*x+13*x^2).

a(n) = 3*A162557(n). - R. J. Mathar, Jun 14 2016

E.g.f.: (1/2)*exp(4*x)*(6*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x)). - G. C. Greubel, Jul 25 2017