Sum over k of n Choose k by Fibonacci Number with index m+k

Theorem

 * $\displaystyle \sum_{k \mathop \ge 0} \binom n k F_{m + k} = F_{m + 2 n}$

where:
 * $\dbinom n k$ denotes a binomial coefficient
 * $F_n$ denotes the $n$th Fibonacci number.

Proof
From Sum over k of n Choose k by Fibonacci t to the k by Fibonacci t-1 to the n-k by Fibonacci m+k:
 * $(1): \quad: \displaystyle \sum_{k \mathop \ge 0} \binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$

Letting $t = 2$ in $(1)$: