IL-GILLIS 2016 Problem 5

The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$ .

Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$ , which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$ .