I was manipulating Fibonacci numbers defined by :
- $F_0=0$ and $F_1=1$
- $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$
Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$:
$F_{n^2}=\sum_{k=0}^n C_n^kF_{n-k}F_{n-1}^{k}F_n^{n-k}$
with $C_n^k=\frac{n!}{k!(n-k)!}$
Questions : Is there a similar formula for the numbers $F_{n^3},\cdots,F_{n^p}$ ? (I'm just curious)
Thank you for your help.