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%I #14 Sep 08 2022 08:45:40
%S 1,16,197,2208,23705,249008,2585533,26677056,274286449,2814636880,
%T 28851289589,295557057504,3026686834313,30989122956272,
%U 317251444075885,3247664850794112,33244802412228577,340304612398804624,3483430456059387941,35656915165420734240
%N a(n) = ((8 + sqrt(5))^n - (8 - sqrt(5))^n)/(2*sqrt(5)).
%C Sixth binomial transform of A048879.
%C lim_{n -> infinity} a(n)/a(n-1) = 8 + sqrt(5) = 10.236067977499789696....
%H G. C. Greubel, <a href="/A153885/b153885.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, -59).
%F From _Philippe Deléham_, Jan 03 2009: (Start)
%F a(n) = 16*a(n-1) - 59*a(n-2) for n>1, with a(0)=0, a(1)=1.
%F G.f.: x/(1 - 16*x + 59*x^2). (End)
%t Join[{a=1,b=16},Table[c=16*b-59*a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2011*)
%t LinearRecurrence[{16, -59}, {1, 16}, 25] (* or *) Table[((8 + sqrt(5))^n - (8 - 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:=[ ((8+r)^n-(8-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; # _Klaus Brockhaus_, Jan 04 2009
%o (Magma) I:=[1,16]; [n le 2 select I[n] else 16*Self(n-1)-59*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Sep 01 2016
%Y Cf. A002163 (decimal expansion of sqrt(5)), A048879.
%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
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