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%I #16 Sep 08 2022 08:45:46
%S 0,1,9,134,2331,41635,746604,13395941,240376941,4313380114,
%T 77400441855,1388894512391,24922700621784,447219716262409,
%U 8025032191009041,144003359719040030,2584035442744223139,46368634609657371691,832051387531037141316,14930556340948876798829
%N Sum of the cubes of the first n odd-indexed Fibonacci numbers.
%C Natural bilateral extension (brackets mark index 0): ..., -41635, -2331, -134, -9, -1, [0], 1, 9, 134, 2331, 41635, ... This is (-A163200)-reversed followed by A163200, without repeating the 0. That is, a(-n) = -a(n). Thus a(n) is an odd function of n.
%H G. C. Greubel, <a href="/A163200/b163200.txt">Table of n, a(n) for n = 0..500</a>
%H K. Subba Rao, <a href="http://www.jstor.org/stable/2307147">Some properties of Fibonacci numbers</a>, Amer. Math. Monthly, 60(10):680-684, Dec. 1953. See page 682.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (21,-56,21,-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} F(2k-1)^3.
%F a(n) = (1/20)*(F(6*n) + 12*F(2*n)).
%F a(n) = (1/4)*(F(2n)^3 + 3*F(2n)). (K. Subba Rao)
%F a(n) = (1/20)*F(2n)*(L(4n) + 13).
%F a(n) = (1/4)*F(2n)*(F(2n)^2 + 3).
%F a(n) - 21*a(n-1) + 56*a(n-2) - 21*a(n-3) + a(n-4) = 0.
%F G.f.: (x - 12*x^2 + x^3)/(1 - 21*x + 56*x^2 - 21*x^3 + x^4) = x*(1 - 12*x + x^2)/((1 - 3*x + x^2 )*(1 - 18*x + x^2)).
%t a[n_Integer] := If[ n >= 0, Sum[ Fibonacci[2k-1]^3, {k, 1, n} ], -Sum[ Fibonacci[-2k+1]^3, {k, 1, -n} ] ]
%t LinearRecurrence[{21,-56,21,-1}, {0,1,9,134}, 50] (* or *) Table[(1/20)*(Fibonacci[6*n] + 12*fibonacci[2*n]),{n,0,25}] (* _G. C. Greubel_, Dec 09 2016 *)
%t Join[{0},Accumulate[Fibonacci[Range[1,41,2]]^3]] (* _Harvey P. Dale_, Jul 20 2021 *)
%o (PARI) concat([0],Vec(x*(1 - 12*x + x^2)/((1 - 3*x + x^2 )*(1 - 18*x + x^2)) + O(x^50))) \\ _G. C. Greubel_, Dec 09 2016
%o (Magma) [(1/4)*Fibonacci(2*n)*(Fibonacci(2*n)^2+3): n in [0..20]]; // _Vincenzo Librandi_, Dec 10 2016
%Y Cf. A005968, A163198, A163201, A163202.
%K nonn,easy
%O 0,3
%A _Stuart Clary_, Jul 24 2009
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