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%I #14 Sep 08 2022 08:45:46
%S 0,-1,26,-486,8775,-157600,2828384,-50754249,910750554,-16342762150,
%T 293258984975,-5262319011456,94428483336576,-1694450381348881,
%U 30405678381733850,-545607760491930150,9790534010478427479,-175684004428133950624,3152521545695969823584,-56569703818099420107225
%N Alternating sum of the cubes of the first n even-indexed Fibonacci numbers.
%C Natural bilateral extension (brackets mark index 0): ..., 8775, -486, 26, -1, 0, [0], -1, 26, -486, 8775, -157600, ... This is A163201-reversed followed by A163201. That is, a(-n) = a(n-1).
%H G. C. Greubel, <a href="/A163201/b163201.txt">Table of n, a(n) for n = 0..500</a>
%H Stuart Clary and Paul D. Hemenway, <a href="http://dx.doi.org/10.1007/978-94-011-2058-6_12">On sums of cubes of Fibonacci numbers</a>, Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 1992), 123-136, Kluwer Acad. Publ., 1993. For the factored closed form, let alpha equal the imaginary unit in Equation (21).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-20,-35,35,20,1).
%F Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
%F a(n) = Sum_{k=1..n} (-1)^k F(2k)^3.
%F a(n) = (1/50)*(L(6n+3) - 6 L(2n+1) + 2) if n is even.
%F a(n) = -(1/50)*(L(6n+3) - 6 L(2n+1) - 2) if n is odd.
%F a(n) = (1/2) * F(n)^2 * F(n+1)^2 * (L(2n+1) + 2) if n is even.
%F a(n) = -(1/2) * F(n)^2 * F(n+1)^2 * (L(2n+1) - 2) if n is odd.
%F a(n) + 21*a(n-1) + 56*a(n-2) + 21*a(n-3) + a(n-4) = 4.
%F a(n) + 20*a(n-1) + 35*a(n-2) - 35*a(n-3) - 20*a(n-4) - a(n-5) = 0.
%F G.f.: (-x + 6*x^2 - x^3)/(1 + 20*x + 35*x^2 - 35*x^3 - 20*x^4 - x^5) = -x*(1 - 6*x + x^2)/((1 - x)*(1 + 3*x + x^2)*(1 + 18*x + x^2)).
%t a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[2k]^3, {k, 1, n} ], -Sum[ (-1)^k Fibonacci[-2k]^3, {k, 1, -n - 1} ] ]
%t LinearRecurrence[{-20, -35, 35, 20, 1}, {0, -1, 26, -486, 8775}, 50] (* or *) Table[(-1)^n*(1/50)*(LucasL[6 n + 3] - 6 LucasL[2 n + 1] + 2*(-1)^n), {n, 0, 25}] (* _G. C. Greubel_, Dec 10 2016 *)
%o (PARI) concat([0], Vec(-x*(1 - 6*x + x^2)/((1 - x)*(1 + 3*x + x^2)*(1 + 18*x + x^2)) + O(x^50))) \\ _G. C. Greubel_, Dec 10 2016
%o (Magma) [(-1)^n*(1/50)*(Lucas(6*n+3)-6*Lucas(2*n+1)+2*(-1)^n): n in [0..20]]; // _Vincenzo Librandi_, Dec 10 2016
%Y Cf. A119284, A163198, A163200, A163202.
%K sign,easy
%O 0,3
%A _Stuart Clary_, Jul 24 2009
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