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

1,2

Comments

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

Links

Table of n, a(n) for n=1..24.

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

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

Example

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

Mathematica

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

Crossrefs

Cf. A030192, A030523.

Sequence in context: A347029 A141223 A289783 * A026388 A242509 A057969

Adjacent sequences: A140763 A140764 A140765 * A140767 A140768 A140769

Keyword

nonn,easy

Author

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

Extensions

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

More terms from Harvey P. Dale, Oct 01 2014

Status

approved