%I #36 Sep 08 2022 08:45:40
%S 1,18,248,3096,36880,428544,4910912,55827072,631657984,7126986240,
%T 80279745536,903384465408,10159659716608,114216655527936,
%U 1283765661040640,14427316078608384,162125499175862272,1821782963191283712,20470555400077574144
%N a(n) = ((9 + sqrt(5))^n - (9 - sqrt(5))^n)/(2*sqrt(5)).
%C lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(5) = 11.236067977499789696....
%H G. C. Greubel, <a href="/A153886/b153886.txt">Table of n, a(n) for n = 1..950</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18, -76).
%F From _Philippe Deléham_ and _Klaus Brockhaus_, Jan 03 2009: (Start)
%F a(n) = 18*a(n-1) - 76*a(n-2) for n>1, with a(0)=0, a(1)=1.
%F G.f.: x/(1 - 18*x + 76*x^2). (End)
%F E.g.f.: sinh(sqrt(5)*x)*exp(9*x)/sqrt(5). - _Ilya Gutkovskiy_, Sep 01 2016
%t Join[{a=1,b=18},Table[c=18*b-76*a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 09 2011 *)
%t LinearRecurrence[{18,-76},{1,18},20] (* _Harvey P. Dale_, Jun 06 2011 *)
%t Table[Simplify[((9 + Sqrt[5])^n - (9 - Sqrt[5])^n)/(2 Sqrt[5])], {n, 1, 25}] (* _G. C. Greubel_, Aug 31 2016 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jan 04 2009
%o (Magma) I:=[1,18]; [n le 2 select I[n] else 18*Self(n-1)-76*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 01 2016
%o (PARI) Vec(x/(1-18*x+76*x^2) + O(x^99)) \\ _Altug Alkan_, Sep 01 2016
%Y Cf. A002163 (decimal expansion of sqrt(5)).
%K nonn
%O 1,2
%A Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009
%E Extended beyond a(7) by _Klaus Brockhaus_, Jan 04 2009
%E Edited by _Klaus Brockhaus_, Oct 11 2009