A147961 a(n) = ((6+sqrt(3))^n + (6-sqrt(3))^n)/2.

1, 6, 39, 270, 1953, 14526, 109863, 838998, 6442497, 49623030, 382873959, 2956927518, 22848289569, 176600866734, 1365216845031, 10554773538150, 81605126571777, 630953992102374, 4878478728359847, 37720263000939822, 291653357975402913, 2255071616673820830, 17436298586897553831

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Keywords

• nonn
• easy

Examples

			a(3)=270
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-33).

Crossrefs

Cf. A098158.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((6+r3)^n+(6-r3)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

Mathematica

CoefficientList[Series[(1-6x)/(1-12x+33x^2),{x,0,30}],x] (* or *) LinearRecurrence[{12,-33},{1,6},30] (* Harvey P. Dale, Jul 30 2021 *)

Formula

From Philippe Deléham, Nov 19 2008: (Start)

a(n) = 12*a(n-1) - 33*a(n-2) for n > 1, with a(0)=1, a(1)=6.

G.f.: (1-6x)/(1-12x+33x^2).

a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*3^(n-k))/6^n. (End)

E.g.f.: exp(6*x)*cosh(sqrt(3)*x). - Stefano Spezia, Apr 23 2025

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008