OFFSET
0,3
COMMENTS
This is one of the triple Fibonacci sums that were considered by Subba Rao (1953).
Taking any of the given closed-form expressions for a(n) with Fibonacci numbers, one can extend a(n) to negative indices by using the property F(-n)=(-1)^(n+1). This gives a(-n)=a(n-1).
LINKS
K. Subba Rao, Some properties of Fibonacci numbers, The American Mathematical Monthly, 60(10):680-684, December 1953.
Index entries for linear recurrences with constant coefficients, signature (22,-77,77,-22,1).
FORMULA
a(n) = (F(2n+1)^3 + F(2n+1) - 2)/8.
a(n) = (F(6*n+3)+8*F(2*n+1)-10)/40.
a(n) = 22*a(n-1) - 77*a(n-2) + 77*a(n-3) - 22*a(n-4) + a(n-5).
G.f.: x*(1 - 6*x + x^2)/((1 - x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)).
MATHEMATICA
LinearRecurrence[{22, -77, 77, -22, 1}, {0, 1, 16, 276, 4917}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Hans J. H. Tuenter, Jun 19 2023
STATUS
approved