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%I #44 Dec 14 2022 06:12:46
%S 1,4,33,268,2177,17684,143649,1166876,9478657,76996132,625447713,
%T 5080577836,41270070401,335241141044,2723199198753,22120834731068,
%U 179689877047297,1459639851109444,11856808685922849,96314109338492236,782369683393860737,6355271576489378132,51624542295308885793
%N a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.
%H G. C. Greubel, <a href="/A088317/b088317.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,1).
%F a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.
%F a(n) = A086594(n)/2.
%F Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).
%F From _Paul Barry_, Nov 15 2003: (Start)
%F E.g.f.: exp(4*x)*cosh(sqrt(17)*x).
%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).
%F a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
%F a(n) = A041024(n-1), n>0. - _R. J. Mathar_, Sep 11 2008
%F G.f.: (1-4*x)/(1-8*x-x^2). - _Philippe Deléham_, Nov 16 2008 and Nov 20 2008
%F a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - _Harvey P. Dale_, May 07 2012
%t LinearRecurrence[{8,1},{1,4},30] (* or *) With[{c=Sqrt[17]},Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)),{n,30}]] (* _Harvey P. Dale_, May 07 2012 *)
%o (Maxima)
%o a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
%o makelist(A088317(n),n,0,20); /* _Martin Ettl_, Nov 12 2012 */
%o (Magma) [n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // _G. C. Greubel_, Dec 13 2022
%o (SageMath)
%o A088317=BinaryRecurrenceSequence(8,1,1,4)
%o [A088317(n) for n in range(31)] # _G. C. Greubel_, Dec 13 2022
%Y Cf. A002018, A002190, A013192, A028576, A041024, A041027.
%Y Cf. A053120, A058153, A058155, A072754, A075132.
%K nonn,easy
%O 0,2
%A Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003