A179237 a(0) = 1, a(1) = 2; a(n+1) = 6*a(n) + a(n-1) for n>1.

1, 2, 13, 80, 493, 3038, 18721, 115364, 710905, 4380794, 26995669, 166354808, 1025124517, 6317101910, 38927735977, 239883517772, 1478228842609, 9109256573426, 56133768283165, 345911866272416, 2131604965917661, 13135541661778382, 80944854936587953

Offset: 0

Author

Gary W. Adamson, Jul 04 2010

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) converges to 1/(sqrt(10) - 3) = 6.16227766017... = A176398.

Examples

			a(5) = 3038 = 6*a(5) + a(4) = 6*493 + 80.
a(5) = term (1,1) in M^5 where M^5 = [3038, 4215, 4215, 5848].
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,1).

Programs

Magma

I:=[1,2]; [n le 2 select I[n] else 6*Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 13 2015

Mathematica

CoefficientList[Series[(-1 + 4 x)/(-1 + 6 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 13 2015 *)

PARI

Vec((-1+4*x)/(-1+6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 13 2015

Formula

Let M = the 2x2 matrix [2,3; 3,4]. a(n) = term (1,1) in M^n.

G.f.: ( -1+4*x ) / ( -1+6*x+x^2 ). a(n) = A005668(n) + A015451(n). - R. J. Mathar, Jul 06 2012

a(n) = ((3-sqrt(10))^n*(1+sqrt(10))+(-1+sqrt(10))*(3+sqrt(10))^n)/(2*sqrt(10)). - Colin Barker, Oct 13 2015

a(n) = Sum_{k=0..n-2} A168561(n-2,k)*6^k + 2 * Sum_{k=0..n-1} A168561(n-1,k)*6^k, n>0. - R. J. Mathar, Feb 14 2024

a(n) = A005668(n+1) - 4*A005668(n). - R. J. Mathar, Feb 14 2024

Extensions

Corrected by R. J. Mathar, Jul 06 2012