A163200 Sum of the cubes of the first n odd-indexed Fibonacci numbers.

0, 1, 9, 134, 2331, 41635, 746604, 13395941, 240376941, 4313380114, 77400441855, 1388894512391, 24922700621784, 447219716262409, 8025032191009041, 144003359719040030, 2584035442744223139, 46368634609657371691, 832051387531037141316, 14930556340948876798829

Offset

0,3

Comments

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

K. Subba Rao, Some properties of Fibonacci numbers, Amer. Math. Monthly, 60(10):680-684, Dec. 1953. See page 682.

Index entries for linear recurrences with constant coefficients, signature (21,-56,21,-1).

Formula

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).

a(n) = Sum_{k=1..n} F(2k-1)^3.

a(n) = (1/20)*(F(6*n) + 12*F(2*n)).

a(n) = (1/4)*(F(2n)^3 + 3*F(2n)). (K. Subba Rao)

a(n) = (1/20)*F(2n)*(L(4n) + 13).

a(n) = (1/4)*F(2n)*(F(2n)^2 + 3).

a(n) - 21*a(n-1) + 56*a(n-2) - 21*a(n-3) + a(n-4) = 0.

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)).

Mathematica

a[n_Integer] := If[ n >= 0, Sum[ Fibonacci[2k-1]^3, {k, 1, n} ], -Sum[ Fibonacci[-2k+1]^3, {k, 1, -n} ] ]
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 *)
Join[{0}, Accumulate[Fibonacci[Range[1, 41, 2]]^3]] (* Harvey P. Dale, Jul 20 2021 *)

Prog

(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
(Magma) [(1/4)*Fibonacci(2*n)*(Fibonacci(2*n)^2+3): n in [0..20]]; // Vincenzo Librandi, Dec 10 2016

Crossrefs

Cf. A005968, A163198, A163201, A163202.

Sequence in context: A268654 A112426 A213688 * A279975 A296171 A167893

Adjacent sequences: A163197 A163198 A163199 * A163201 A163202 A163203

Keyword

nonn,easy

Author

Stuart Clary, Jul 24 2009

Status

approved