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

Theorem

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

where:

$\dbinom n k$ denotes a binomial coefficient

$F_n$ denotes the $n$th Fibonacci number.

$(1): \quad \ds \sum_{k \mathop \ge 0} \binom n k {F_t}^k {F_{t - 1} }^{n - k} F_{m + k} = F_{m + t n}$

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

$\ds \sum_{k \mathop \ge 0} \binom n k {F_2}^k {F_{2 - 1} }^{n - k} F_{m + k}$

$=$

$\ds \sum_{k \mathop \ge 0} \binom n k 1^k 1^{n - k} F_{m + k}$

Definition of Fibonacci Number: $F_1 = 1, F_2 = 1$

$\ds $

$=$

$\ds \sum_{k \mathop \ge 0} \binom n k F_{m + k}$

$\ds $

$=$

$\ds F_{m + 2 n}$

from $(1)$

$\blacksquare$