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 *k*th 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*.