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%I #38 Sep 08 2022 08:45:06
%S 2,3,19,72,311,1293,5434,22767,95471,400248,1678099,7035537,29497106,
%T 123669003,518492539,2173822632,9113930591,38210904933,160202367754,
%U 671661627927,2815996722551,11806298307288,49498878534619,207528127140297
%N a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=2, a(1)=3.
%C Inverse binomial transform of A087130. - _Johannes W. Meijer_, Aug 01 2010
%C Pisano period lengths: 1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12... - _R. J. Mathar_, Aug 10 2012
%C This is the Lucas sequence V(3,-5). - _Bruno Berselli_, Jan 09 2013
%H G. C. Greubel, <a href="/A072263/b072263.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,5).
%F a(n) = 2*A015523(n+1) - 3*A015523(n).
%F a(n) = ((3 + sqrt(29))/2)^n + ((3 - sqrt(29))/2)^n.
%F G.f.: (2-3*x)/(1-3*x-5*x^2). - R. J. Mathar, Feb 06 2010
%F From _Johannes W. Meijer_, Aug 01 2010: (Start)
%F Limit_{k -> Infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2.
%F Limit_{n -> Infinity} (A072263(n)/A015523(n)) = sqrt(29). (End)
%F G.f.: G(0), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 17 2013
%F a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 29*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F a(n) = 5^((n-1)/2)*( 2*sqrt(5)*Fibonacci(n+1, 3/sqrt(5)) - 3*Fibonacci(n, 3/sqrt(5)) ). - _G. C. Greubel_, Jan 14 2020
%e a(5)=5*b(4)+b(6): 1293=5*57+1008.
%p seq(coeff(series((2-3*x)/(1-3*x-5*x^2), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Jan 14 2020
%t LinearRecurrence[{3,5},{2,3},40] (* _Harvey P. Dale_, Nov 23 2018 *)
%o (Sage) [lucas_number2(n,3,-5) for n in range(0, 16)] # _Zerinvary Lajos_, Apr 30 2009
%o (PARI) my(x='x+O('x^40)); Vec((2-3*x)/(1-3*x-5*x^2)) \\ _G. C. Greubel_, Jan 14 2020
%o (Magma) I:=[2,3]; [n le 2 select I[n] else 3*Self(n-1) +5*Self(n-2): n in [1..40]]; // _G. C. Greubel_, Jan 14 2020
%o (GAP) a:=[2,3];; for n in [3..40] do a[n]:=3*a[n-1]+5*a[n-2]; od; a; # _G. C. Greubel_, Jan 14 2020
%Y Cf. A072264, A152187, A197189.
%Y Appears in A179606 and A015523. - _Johannes W. Meijer_, Aug 01 2010
%K nonn,easy
%O 0,1
%A _Miklos Kristof_, Jul 08 2002
%E Offset changed and terms added by _Johannes W. Meijer_, Jul 19 2010
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