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Expansion of (4+3*x+3*x^2+2*x^3-x^5-x^6-x^7)/(1-x^4)^2.
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%I #31 Mar 03 2024 14:27:51

%S 4,3,3,2,8,5,5,3,12,7,7,4,16,9,9,5,20,11,11,6,24,13,13,7,28,15,15,8,

%T 32,17,17,9,36,19,19,10,40,21,21,11,44,23,23,12,48,25,25,13,52,27,27,

%U 14,56,29,29,15,60,31,31,16,64,33,33,17,68,35,35,18,72,37,37,19,76,39,39,20

%N Expansion of (4+3*x+3*x^2+2*x^3-x^5-x^6-x^7)/(1-x^4)^2.

%H G. C. Greubel, <a href="/A117691/b117691.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 2, 0, 0, 0, -1).

%F G.f.: (4+3*x+3*x^2+2*x^3-x^5-x^6-x^7) / ((1-x)*(1+x)*(1+x^2))^2. - _Colin Barker_, Mar 15 2013

%F a(n) = 2*a(n-4) - a(n-8). - _Colin Barker_, Mar 15 2013

%t F= Table[Abs[Coefficient[ExpandAll[(x -(m+I*Sqrt[2*m+1])/(m+1))*(x -( m-I*Sqrt[2*m+1])/(m+1))], x]], {m,100}];

%t Table[{Numerator[F[[n]]], Denominator[F[[n]]]}, {n,2,100}]//Flatten

%t (* Additional programs *)

%t t= {4,3,3,2,8,5,5,3}; Do[AppendTo[t, 2*t[[-4]] - t[[-8]]], {100}]; t (* _T. D. Noe_, Mar 17 2013 *)

%t LinearRecurrence[{0,0,0,2,0,0,0,-1},{4,3,3,2,8,5,5,3},80] (* _Harvey P. Dale_, Jan 30 2021 *)

%o (Magma) I:=[4,3,3,2,8,5,5,3]; [n le 8 select I[n] else 2*Self(n-4) - Self(n-8): n in [1..80]]; // _G. C. Greubel_, Jul 21 2023

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A117691

%o if (n<8): return (4,3,3,2,8,5,5,3)[n]

%o else: return 2*a(n-4) - a(n-8)

%o [a(n) for n in range(81)] # _G. C. Greubel_, Jul 21 2023

%K nonn,easy

%O 0,1

%A _Roger L. Bagula_, Apr 12 2006

%E New name from _Colin Barker_, Mar 17 2013