A153597 a(n) = ((6 + sqrt(3))^n - (6 - sqrt(3))^n)/(2*sqrt(3)).

1, 12, 111, 936, 7569, 59940, 469503, 3656016, 28378593, 219894588, 1702241487, 13170376440, 101870548209, 787824155988, 6092161780959, 47107744223904, 364251591915201, 2816463543593580, 21777259989921327, 168383822940467784

Offset: 1

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Keywords

• nonn

Comments

Fourth binomial transform of A055845.

lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(3) = 7.73205080756887729....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-33).

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)), A055845.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008

Magma

I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-33*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Mathematica

LinearRecurrence[{12,-33},{1, 12},25] (* G. C. Greubel, Aug 22 2016 *)

Sage

[lucas_number1(n,12,33) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

Formula

G.f.: x/(1 - 12*x + 33*x^2). - Klaus Brockhaus, Dec 31 2008, (corrected Oct 11 2009)

a(n) = 12*a(n-1) - 33*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

E.g.f.: sinh(sqrt(3)*x)*exp(6*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016

Extensions

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009