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A163202 Alternating sum of the cubes of the first n odd-indexed Fibonacci numbers. 5
0, -1, 7, -118, 2079, -37225, 667744, -11981593, 214999407, -3858003766, 69229057975, -1242265012561, 22291541096832, -400005474543793, 7177807000202839, -128800520527828150, 2311231562497354959, -41473367604415793593, 744209385316963976032, -13354295568100875681481 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Natural bilateral extension (brackets mark index 0): ..., -37225, 2079, -118, 7, -1, [0], -1, 7, -118, 2079, -37225, ... This is A163202-reversed followed by A163202, without repeating the 0. That is, A163202(-n) = A163202(n). Thus A163202(n) is an even function of n.

LINKS

Table of n, a(n) for n=0..19.

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} (-1)^k F(2k-1)^3

Closed form: (1/50)(L(6n) + 6 L(2n) - 14) if n is even; -(1/50)(L(6n) + 6 L(2n) + 14) if n is odd

Factored closed form: 1/10) F(n)^2 (L(4 n) + 2 L(2n) + 9) if n is even; -(1/10) F(n)^2 (L(4 n) - 2 L(2n) + 9) if n is odd

Recurrence: a(n) + 21 a(n-1) + 56 a(n-2) + 21 a(n-3) + a(n-4) = -28

Recurrence: a(n) + 20 a(n-1) + 35 a(n-2) - 35 a(n-3) - 20 a(n-4) - a(n-5) = 0

G.f.: A(x) = (-x - 13 x^2 - 13 x^3 - x^4)/(1 + 20 x + 35 x^2 - 35 x^3 - 20 x^4 - x^5) = -x(1 + x)(1 + 12 x +x^2)/((1 - x)(1 + 3 x + x^2)(1 + 18 x + x^2))

a(-n) = a(n). - Michael Somos Aug 11 2009

EXAMPLE

-x + 7*x^2 - 118*x^3 + 2079*x^4 - 37225*x^5 + 667744*x^6 - 11981593*x^7 + ... - Michael Somos Aug 11 2009

MATHEMATICA

a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[2k-1]^3, {k, 1, n} ], Sum[ (-1)^k Fibonacci[-2k+1]^3, {k, 1, -n} ] ]

Join[{0}, Accumulate[Times@@@Partition[Riffle[Take[Fibonacci[Range[41]], {1, -1, 2}]^3, {-1, 1}], 2]]] (* or *) LinearRecurrence[{-20, -35, 35, 20, 1}, {0, -1, 7, -118, 2079}, 20] (* Harvey P. Dale, Feb 19 2012 *)

PROG

(PARI) {a(n) = ((-1)^n * (fibonacci(6*n) / 2 + fibonacci(6*n - 1) + 3*fibonacci(2*n - 1) + 3*fibonacci(2*n + 1)) - 7) / 25} /* Michael Somos Aug 11 2009 */

CROSSREFS

Cf. A163198, A163200, A163201, A119284

Sequence in context: A009696 A218049 A097202 * A213112 A076283 A057769

Adjacent sequences:  A163199 A163200 A163201 * A163203 A163204 A163205

KEYWORD

sign,easy

AUTHOR

Stuart Clary, Jul 24, 2009

STATUS

approved

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Last modified April 16 21:07 EDT 2014. Contains 240627 sequences.