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

Author

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

Keywords

• nonn

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).

Crossrefs

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

Programs

Magma

Z:=PolynomialRing(Integers()); N:=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

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

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 *)

Sage

[lucas_number1(n,12,30) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

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)

Extensions

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

Edited by Klaus Brockhaus, Oct 06 2009