A154237 a(n) = ( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)).

1, 12, 114, 1008, 8676, 73872, 626184, 5298048, 44791056, 378551232, 3198883104, 27030060288, 228394230336, 1929828955392, 16306120554624, 137778577993728, 1164159319286016, 9836554491620352, 83113874320863744, 702269857101754368

Offset

1,2

Comments

Fifth binomial transform of A002533 without initial term 1. Sixth binomial transform of 1 followed by A056452.

Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(6) = 8.4494897427....

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (12, -30).

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 12*a(n-1) - 30*a(n-2) for n > 1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 12*x + 30*x^2). (End)

Mathematica

Join[{a=1, b=12}, Table[c=12*b-30*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011*)
LinearRecurrence[{12, -30}, {1, 12}, 25] (* or *) Table[( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)), {n, 1, 25}] (* G. C. Greubel, Sep 07 2016 *)

Prog

(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-6); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009
(Sage) [lucas_number1(n, 12, 30) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009
(Magma) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-30*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Sep 07 2016

Crossrefs

Cf. A010464 (decimal expansion of square root of 6), A002533, A056452.

Sequence in context: A125400 A378504 A331515 * A006635 A062386 A080602

Adjacent sequences: A154234 A154235 A154236 * A154238 A154239 A154240

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

Status

approved