A162273 a(n) = ((2+sqrt(3))*(3+sqrt(3))^n + (2-sqrt(3))*(3-sqrt(3))^n)/2.

2, 9, 42, 198, 936, 4428, 20952, 99144, 469152, 2220048, 10505376, 49711968, 235239552, 1113165504, 5267555712, 24926341248, 117952713216, 558158231808, 2641233111552, 12498449278464, 59143297001472, 279869086338048

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A001075 without initial term 1, inverse binomial transform of A162274.

The INVERTi transform yields A007051 without A007051(0). - R. J. Mathar, Jul 07 2009

Links

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

Crossrefs

Cf. A001075, A162274.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((2+r)*(3+r)^n+(2-r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 05 2009

Maple

seq(simplify(((2+sqrt(3))*(3+sqrt(3))^n+(2-sqrt(3))*(3-sqrt(3))^n)*1/2), n = 0 .. 22); # Emeric Deutsch, Jul 11 2009

Mathematica

LinearRecurrence[{6,-6},{2,9},30] (* Harvey P. Dale, Dec 17 2019 *)

Formula

a(n) = 6*a(n-1) - 6*a(n-2) for n > 1; a(0) = 2, a(1) = 9.

G.f.: (2-3*x)/(1-6*x+6*x^2).

a(n) = 2*A030192-3*A030192(n-1). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by R. J. Mathar and Klaus Brockhaus, Jul 05 2009