How is this identity for Fibonacci numbers called?

Question

In the course of proving another identity, I've found that

$$F_n \equiv F_kF_{n-k+1} + F_{k-1}F_{n-k}$$

…for all corresponding n and k. However, this (or something similar) has assuredly already been named. What's the above referred to as?

Answer 1

It's advisable to check Wolfram Mathworld first… this identity has been given by Honsberger (1985, p. 107), presented as formula 55 on the linked page.