On Sat, Jul 13, 2013 at 11:03 AM, Bill Gosper <billgosper@gmail.com> wrote:

<< Was anybody surprised by Fibonacci[6*n]*Fibonacci[n]/ Fibonacci[3*n]/Fibonacci[2*n] - Fibonacci[3*n]/Fibonacci[n] ? NeilB found a short Mma proof. >>

Fred> I suppose it is a little unexpected that the answer is not exactly zero ... We still have the nice little puzzle of how NeilB got Mma to say -2*(-1)^n.

He then quickly found In[149]:= FunctionExpand[ Fibonacci[10*n]*Fibonacci[n]/Fibonacci[2*n]/Fibonacci[5*n] - Fibonacci[8*n]/Fibonacci[4*n] + (-1)^n*

Fibonacci[4*n]/Fibonacci[2*n], n \[Element] Integers]

Out[149]= 1

FullSimplify instead of FunctionExpand also works, but slower.

Oddly, discharging both barrels at once,

In[146]:= FullSimplify[

FunctionExpand[ Fibonacci[10*n]*Fibonacci[n]/Fibonacci[2*n]/Fibonacci[5*n] - Fibonacci[8*n]/Fibonacci[4*n] + (-1)^n* Fibonacci[4*n]/Fibonacci[2*n]], n \[Element] Integers]

Out[146]= ((-(-1)^n)*4^(5*n) + (1 + Sqrt[5])^(10*n))/

(-(-4)^(5*n) + (1 + Sqrt[5])^(10*n))

A point from which Mma seems unable to reach 1. ?

Taking a page from NeilB's book, if you already know or suspect 1, In[162]:= FullSimplify[%146 - 1, n \[Element] Integers] + 1 Out[162]= 1 FullSimplify is a crapshoot. Similarly, it can confirm what you already suspect of In[172]:= FullSimplify[Fibonacci[6*n]*Fibonacci[n]/Fibonacci[3*n]/Fibonacci[2*n] - Fibonacci[3*n]/Fibonacci[n] + 2*(-1)^n, n \[Element] Integers] - 2*(-1)^n Out[172]= -2 (-1)^n Neil's solution was *slightly* more respectable. --rwg