cp's OEIS Frontend

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

%I A154237 #23 Sep 08 2022 08:45:40
%S A154237 1,12,114,1008,8676,73872,626184,5298048,44791056,378551232,
%T A154237 3198883104,27030060288,228394230336,1929828955392,16306120554624,
%U A154237 137778577993728,1164159319286016,9836554491620352,83113874320863744,702269857101754368
%N A154237 a(n) = ( (6 + sqrt(6))^n - (6 - sqrt(6))^n )/(2*sqrt(6)).
%C A154237 Fifth binomial transform of A002533 without initial term 1. Sixth binomial transform of 1 followed by A056452.
%C A154237 Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(6) = 8.4494897427....
%H A154237 G. C. Greubel, <a href="/A154237/b154237.txt">Table of n, a(n) for n = 1..1000</a>
%H A154237 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12, -30).
%F A154237 From _Philippe Deléham_, Jan 06 2009: (Start)
%F A154237 a(n) = 12*a(n-1) - 30*a(n-2) for n > 1, with a(0)=0, a(1)=1.
%F A154237 G.f.: x/(1 - 12*x + 30*x^2). (End)
%t A154237 Join[{a=1,b=12},Table[c=12*b-30*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2011*)
%t A154237 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 *)
%o A154237 (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
%o A154237 (Sage) [lucas_number1(n,12,30) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 27 2009
%o A154237 (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
%Y A154237 Cf. A010464 (decimal expansion of square root of 6), A002533, A056452.
%K A154237 nonn
%O A154237 1,2
%A A154237 Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%E A154237 Extended beyond a(7) by _Klaus Brockhaus_, Jan 07 2009
%E A154237 Edited by _Klaus Brockhaus_, Oct 06 2009