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A080042 a(n) = 4*a(n-1)+3*a(n-2) for n>1, a(0)=2, a(1)=4.

%I A080042 #24 Jun 17 2023 07:25:58
%S A080042 2,4,22,100,466,2164,10054,46708,216994,1008100,4683382,21757828,
%T A080042 101081458,469599316,2181641638,10135364500,47086382914,218751625156,
%U A080042 1016265649366,4721317472932,21934066839826,101900219778100
%N A080042 a(n) = 4*a(n-1)+3*a(n-2) for n>1, a(0)=2, a(1)=4.
%C A080042 This is the Lucas sequence V(4,-3). [_Bruno Berselli_, Jan 09 2013]
%H A080042 Vincenzo Librandi, <a href="/A080042/b080042.txt">Table of n, a(n) for n = 0..300</a>
%H A080042 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.
%H A080042 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, 3).
%F A080042 G.f.: (2-4*x)/(1-4*x-3*x^2).
%F A080042 a(n) = (2+sqrt(7))^n+(2-sqrt(7))^n.
%F A080042 G.f.: G(0)/x -2/x, where G(k)= 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 03 2013
%F A080042 a(n) = [x^n] ( (1 + 4*x + sqrt(1 + 8*x + 28*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%t A080042 CoefficientList[Series[(2 - 4 t)/(1 - 4 t - 3 t^2), {t, 0, 25}], t]
%o A080042 (Sage) [lucas_number2(n,4,-3) for n in range(0, 22)] # _Zerinvary Lajos_, May 14 2009
%Y A080042 Cf. A015530: Lucas sequence U(4,-3).
%K A080042 nonn,easy
%O A080042 0,1
%A A080042 Mario Catalani (mario.catalani(AT)unito.it), Jan 21 2003