If $f_{(1)} =1=f_{(2)} $ and $f_{(n)} =f_ {(n-1)} +f_ {(n-2)} $ how I prove that $\displaystyle \sum_ {k=1} ^n f_{(k)} = \sum_ {k=1} ^ {\frac {n} {2}} \bigg (\sum_ {p=1} ^ {k-1} \big (\binom {k+p} {2p+1} + \binom {k+p-1} {2p} \big) \bigg) $?
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