A140766 a(n) = 6*a(n-1) - 6*a(n-2).

1, 5, 24, 114, 540, 2556, 12096, 57240, 270864, 1281744, 6065280, 28701216, 135815616, 642686400, 3041224704, 14391229824, 68100030720, 322252805376, 1524916647936, 7215983055360, 34146398444544, 161582492335104, 764616563343360, 3618204426049536

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, May 28 2008

Keywords

• nonn
• easy

Comments

Companion sequence = A030192 beginning (1, 6, 30, 144, 684,...).

Examples

			a(5) = 540 = 6*a(4) - 6*a(3) = 6*(114) - 6*24.
a(5) = 540 = term (1,1) of X^5.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-6).

Crossrefs

Cf. A030192, A030523.

Programs

Mathematica

LinearRecurrence[{6,-6},{1,5},30] (* Harvey P. Dale, Oct 01 2014 *)

Formula

a(n) = 6*a(n-1) - 6*a(n-2); a(1) = 1, a(2) = 5.

Term (1,1) of M^n, where M = the 3x3 matrix [1,1,1; 1,2,1; 3,1,3].

From R. J. Mathar, May 31 2008: (Start)

O.g.f.: x*(1 - x)/(1 - 6*x + 6*x^2).

a(n) = A030192(n-1) - A030192(n-2). (End)

a(n) = Sum_{1<=k<=n} A030523(n,k)*2^(k-1). - Philippe Deléham, Feb 19 2013

a(n) = (sqrt(3)/36)*((3 + sqrt(3))^(n+1) - (3 - sqrt(3))^(n+1)). - Taras Goy, Jan 03 2025

E.g.f.: (exp(3*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 1)/6. - Stefano Spezia, Jan 04 2025

Extensions

More terms from R. J. Mathar, May 31 2008

More terms from Harvey P. Dale, Oct 01 2014