Question: Let $F_m$ be the mth Fibonacci number given by $F_1=F_2=1$ and $F_{m+2} = F_m + F_{m+1}$ for all $m\ge1$.   Show that  $\sum_m(C_{n,k})=F_{m+1}$ ;  here the sum is over all pairs of integers $n\ge k\ge0$ with $n+k=m$.  My Approach: $F_3=2,F_4=3,F_5=5$;  Now for $m=4$: $F_5=5$ and $(n,k)$ for $m=4$ are $(4,0),(3,1),(2,2)$, so $\sum_{n+k=4} C_{n,k}=C_{4,0} + C_{3,1} + C_{2,2} = 1+3+1 = 5 $.  Next how can i proceed??
The binomial sum   $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$   satisfies the Fibonacci relation.   I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any hints or suggestions?
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