A083695 a(n) = 2*a(n-1) + 5*a(n-2), with a(0) = 0, a(1) = 3.

0, 3, 6, 27, 84, 303, 1026, 3567, 12264, 42363, 146046, 503907, 1738044, 5995623, 20681466, 71341047, 246089424, 848884083, 2928215286, 10100850987, 34842778404, 120189811743, 414593515506, 1430136089727, 4933239756984, 17017159962603, 58700518710126

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), May 03 2003

Keywords

• easy
• nonn

Comments

A002533(n)/a(n) converges to sqrt(2/3).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,5).

Programs

Magma

I:=[0,3]; [n le 2 select I[n] else 2*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 07 2013

Mathematica

CoefficientList[Series[3x/(1-2x-5x^2), {x, 0, 25}], x]
LinearRecurrence[{2, 5}, {0, 3}, 27] (* Arkadiusz Wesolowski, May 21 2013 *)

Formula

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

a(n) = 3[(1+sqrt(6))^n-(1-sqrt(6))^n]/(2*sqrt(6)).

a(n) = 3*A002533(n-1) + a(n-1).

G.f.: 1/Q(0) -1, where Q(k) = 1 + 5*x^2 - (3*k+4)*x + x*(3*k+1 - 5*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013