A163202 Alternating sum of the cubes of the first n odd-indexed Fibonacci numbers.

0, -1, 7, -118, 2079, -37225, 667744, -11981593, 214999407, -3858003766, 69229057975, -1242265012561, 22291541096832, -400005474543793, 7177807000202839, -128800520527828150, 2311231562497354959, -41473367604415793593, 744209385316963976032, -13354295568100875681481

Offset: 0

Author

Stuart Clary, Jul 24 2009

Keywords

• sign
• easy

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, a(-n) = a(n). Thus a(n) is an even function of n.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (-20, -35, 35, 20, 1).

Crossrefs

Cf. A163198, A163200, A163201, A119284.

Programs

Magma

[((-1)^n*(Fibonacci(6*n)/2+Fibonacci(6*n-1)+ 3*Fibonacci(2*n-1)+3*Fibonacci(2*n+1))-7)/25: n in [0..20]]; // Vincenzo Librandi, Dec 19 2016

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 *)
Table[(-1)^n*(1/50)*(LucasL[6 n] + 6 LucasL[2 n] - 14*(-1)^n), {n,0,50}] (* G. C. Greubel, Dec 10 2016 *)

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 */

PARI

concat([0], Vec(-x*(1 + x)*(1 + 12*x +x^2)/((1 - x)*(1 + 3*x + x^2)*(1 + 18*x + x^2)) + O(x^50))) \\ G. C. Greubel, Dec 10 2016

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.

a(n) = (1/50)*(L(6n) + 6 L(2n) - 14) if n is even.

a(n) = -(1/50)*(L(6n) + 6 L(2n) + 14) if n is odd.

a(n) = (1/10)*F(n)^2*(L(4 n) + 2 L(2n) + 9) if n is even.

a(n) = -(1/10)*F(n)^2*(L(4 n) - 2 L(2n) + 9) if n is odd.

a(n) + 21*a(n-1) + 56*a(n-2) + 21*a(n-3) + a(n-4) = -28.

a(n) + 20*a(n-1) + 35*a(n-2) - 35*a(n-3) - 20*a(n-4) - a(n-5) = 0.

G.f.: (-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