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%I A105063 #24 Mar 13 2023 17:30:13
%S A105063 0,0,8,24,560,1616,36984,106664,2440416,7038240,161030504,464417208,
%T A105063 10625572880,30644497520,701126779608,2022072419144,46263741881280,
%U A105063 133426135166016,3052705837384904,8804102848537944,201432321525522416
%N A105063 a(1)=0, a(2)=0, a(3)=8, a(4)=24, a(n) = 32 + 66*a(n-2) - a(n-4) for n > 4.
%C A105063 This sequence has the property 17*a(n)*(a(n) + 1) + 1 is a square.
%H A105063 G. C. Greubel, <a href="/A105063/b105063.txt">Table of n, a(n) for n = 1..1000</a>
%H A105063 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,66,-66,-1,1).
%F A105063 From _R. J. Mathar_, Nov 13 2009: (Start)
%F A105063 a(n) = a(n-1) +66*a(n-2) -66*a(n-3) -a(n-4) +a(n-5).
%F A105063 G.f.: 8*x^3*(1+x)^2/((1-x)*(1+8*x-x^2)*(1-8*x-x^2)). (End)
%F A105063 a(n) = (1/4)*(-32*[n=0] - 2 + i^n*((23 + 11*(-1)^n)*ChebyshevU(n, 4*I) - i*(187 + 89*(-1)^n)*ChebyshevU(n-1, 4*I))). - _G. C. Greubel_, Mar 13 2023
%t A105063 LinearRecurrence[{1,66,-66,-1,1}, {0,0,8,24,560}, 40] (* _G. C. Greubel_, Mar 13 2023 *)
%o A105063 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( 8*x^3*(1+x)^2/((1-x)*(1-66*x^2+x^4)) )); // _G. C. Greubel_, Mar 13 2023
%o A105063 (SageMath)
%o A105063 @CachedFunction
%o A105063 def a(n):
%o A105063     if (n<6): return (0,0,0,8,24,560)[n]
%o A105063     else: return a(n-1) +66*a(n-2) -66*a(n-3) -a(n-4) +a(n-5)
%o A105063 [a(n) for n in range(1,41)] # _G. C. Greubel_, Mar 13 2023
%Y A105063 Cf. A001090, A103200.
%K A105063 nonn,easy
%O A105063 1,3
%A A105063 _Pierre CAMI_, Apr 05 2005
%E A105063 More terms from _R. J. Mathar_, Nov 13 2009

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