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%I #47 Sep 08 2022 08:45:40
%S 1,8,51,304,1769,10200,58603,336224,1927953,11052712,63358307,
%T 363181200,2081791609,11932977272,68400527259,392075513536,
%U 2247397253921,12882196355400,73841406542227,423262699717616,2426163312691977,13906891405206808
%N a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).
%C Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.
%C First differences are in A161728.
%C Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....
%H G. C. Greubel, <a href="/A153594/b153594.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-13).
%F G.f.: x/(1 - 8*x + 13*x^2). - _Klaus Brockhaus_, Dec 31 2008, corrected Oct 11 2009
%F a(n) = 8*a(n-1) - 13*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(4*x)/sqrt(3). - _Ilya Gutkovskiy_, Aug 23 2016
%F a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - _J. Conrad_, Aug 30 2016
%F a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - _J. Conrad_, Sep 03 2016
%t Join[{a=1,b=8},Table[c=8*b-13*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 19 2011 *)
%t LinearRecurrence[{8,-13},{1,8},40] (* _Harvey P. Dale_, Aug 16 2012 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Dec 31 2008
%o (Sage) [lucas_number1(n,8,13) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 23 2009
%o (Magma) I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 23 2016
%o (PARI) a(n)=([0,1; -13,8]^(n-1)*[1;8])[1,1] \\ _Charles R Greathouse IV_, Sep 04 2016
%Y Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.
%K nonn,easy
%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