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%I #58 Sep 08 2022 08:44:40
%S 0,1,4,19,88,409,1900,8827,41008,190513,885076,4111843,19102600,
%T 88745929,412291516,1915403851,8898489952,41340171361,192056155300,
%U 892245135283,4145149007032,19257331433977,89464772757004
%N Expansion of x/(1 - 4*x - 3*x^2).
%C Let b(1)=1, b(k) = floor(b(k-1)) + 3/b(k-1); then for n>1, b(n) = a(n)/a(n-1). - _Benoit Cloitre_, Sep 09 2002
%C In general, x/(1 - a*x - b*x^2) has a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1,k)*b^k*a^(n-2k-1). - _Paul Barry_, Apr 23 2005
%C Pisano period lengths: 1, 2, 1, 4, 24, 2, 21, 4, 3, 24, 40, 4, 84, 42, 24, 8, 288, 6, 18, 24, ... . - _R. J. Mathar_, Aug 10 2012
%C This is the Lucas sequence U(4,-3). - _Bruno Berselli_, Jan 09 2013
%H Vincenzo Librandi, <a href="/A015530/b015530.txt">Table of n, a(n) for n = 0..1000</a>
%H Lucyna Trojnar-Spelina and Iwona Włoch, <a href="https://doi.org/10.1007/s40995-019-00757-7">On Generalized Pell and Pell-Lucas Numbers</a>, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,3).
%F a(n) = 4*a(n-1) + 3*a(n-2).
%F a(n) = (A086901(n+2) - A086901(n+1))/6. - _Ralf Stephan_, Feb 01 2004
%F a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*3^k*4^(n-2k-1). - _Paul Barry_, Apr 23 2005
%F a(n) = ((2+sqrt(7))^n - (2-sqrt(7))^n)/sqrt(28). Offset 1. a(3)=19. - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
%F From _Johannes W. Meijer_, Aug 01 2010: (Start)
%F Limit(a(n+k)/a(k), k=infinity) = A108851(n)+a(n)*sqrt(7).
%F Limit(A108851(n)/a(n), n=infinity) = sqrt(7). (End)
%F G.f.: x*G(0) where G(k)= 1 + (4*x+3*x^2)/(1 - (4*x+3*x^2)/(4*x + 3*x^2 + 1/G(k+1))); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Jul 28 2012
%F G.f.: G(0)*x/(2-4*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 16 2013
%t LinearRecurrence[{4,3},{0,1},30] (* _Vincenzo Librandi_, Jun 19 2012 *)
%o (Sage) [lucas_number1(n,4,-3) for n in range(0, 23)]# _Zerinvary Lajos_, Apr 23 2009
%o (Magma) I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 19 2012
%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-4*x-3*x^2))) \\ _G. C. Greubel_, Jan 24 2018
%Y Appears in A179596, A126473 and A179597. - _Johannes W. Meijer_, Aug 01 2010
%Y Cf. A080042: Lucas sequence V(4,-3).
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
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