Recurrences for even and odd indexed of Fibonacci Numbers
The right-hand side is the convolution of sequences $\{n\}$ and $\{a_n\}$ (the term $0a_n$ is simply omitted). Taking the ordinary generating functions of both sides, we get $$ A(x)-1=\frac{x}{(1-x)^2}A(x), $$ so that $$ A(x)=\frac{(1-x)^2}{1-3x+x^2}=1+\frac{x}{1-3x+x^2}, $$ and thus $$ a_n=F_{2n}, \quad n\ge 1, $$ where the sequence $\{F_n\}$ starts with $F_0=0$, $F_1=1$.