In a previous post, it was pointed out that powers of Fibonacci numbers also obey recurrence relations. For example, it is the case that,
In general, for even kth powers, it takes k+2 consecutive Fibonacci numbers to sum up to zero. However, using Mathematica’s LatticeReduce function which has an integer relations algorithm, I found that if reduced to k+1 terms, then it can still sum up to a constant, though it is now non-zero. Thus,
and so on, with k = 10 summing to . Notice the formulas are palindromic, the same read forwards or backwards.
I was curious if this sequence of constants,
had a generating function. Unfortunately, OEIS didn’t recognize it, so that question is unanswered for now.
Update, May 26, 2012: Jim Cullen found a recurrence relation which is equivalent to the formula,
hence the next constant is C (12) = 53222400. The product of the first p Fibonacci numbers F (n) is called a fibonorial.