OFFSET
0,3
COMMENTS
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).
FORMULA
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
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 *)
Table[(-1)^n*(1/50)*(LucasL[6 n] + 6 LucasL[2 n] - 14*(-1)^n), {n, 0, 50}] (* G. C. Greubel, Dec 10 2016 *)
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 */
(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
(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
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Stuart Clary, Jul 24 2009
STATUS
approved