%I #17 Sep 08 2022 08:45:40

%S 1,12,111,936,7569,59940,469503,3656016,28378593,219894588,1702241487,

%T 13170376440,101870548209,787824155988,6092161780959,47107744223904,

%U 364251591915201,2816463543593580,21777259989921327,168383822940467784

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

%C Fourth binomial transform of A055845.

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

%H G. C. Greubel, <a href="/A153597/b153597.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-33).

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

%F a(n) = 12*a(n-1) - 33*a(n-2) for n>1; a(0)=0, a(1)=1. - _Philippe Deléham_, Jan 01 2009

%F E.g.f.: sinh(sqrt(3)*x)*exp(6*x)/sqrt(3). - _Ilya Gutkovskiy_, Aug 23 2016

%t LinearRecurrence[{12,-33},{1, 12},25] (* _G. C. Greubel_, Aug 22 2016 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=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

%o (Sage) [lucas_number1(n,12,33) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 27 2009

%o (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

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

%K nonn

%O 1,2

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

%E Extended beyond a(7) by _Klaus Brockhaus_, Dec 31 2008

%E Edited by _Klaus Brockhaus_, Oct 11 2009