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%I #20 Sep 08 2022 08:45:40
%S 1,16,195,2144,22409,227760,2277211,22542016,221762385,2173135184,
%T 21242657459,207321273120,2021338264921,19694814578416,
%U 191815399094475,1867662696228224,18181863794888609,176982396248296080,1722624648484532131,16766068204606453216
%N a(n) = ((8 + sqrt(3))^n - (8 - sqrt(3))^n)/(2*sqrt(3)).
%C lim_{n -> infinity} a(n)/a(n-1) = 8 + sqrt(3) = 9.73205080756887729....
%H G. C. Greubel, <a href="/A153599/b153599.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16, -61).
%F G.f.: x/(1 - 16*x + 61*x^2). - _Klaus Brockhaus_, Dec 31 2008, (corrected Oct 11 2009)
%F a(n) = 16*a(n-1) - 61*a(n-2) for n>1; a(0)=0, a(1)=1. - _Philippe Deléham_, Jan 01 2009
%t Join[{a=1,b=16},Table[c=16*b-61*a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2011*)
%t Simplify/@Table[c=Sqrt[3];((8+c)^n-(8-c)^n)/(2c),{n,20}] (* or *) LinearRecurrence[{16,-61},{1,16},20] (* _Harvey P. Dale_, Sep 24 2012 *)
%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((8+r)^n-(8-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Dec 31 2008
%o (Magma) I:=[1,16]; [n le 2 select I[n] else 16*Self(n-1)-61*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Aug 23 2016
%Y Cf. A002194 (decimal expansion of sqrt(3)).
%K nonn
%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