Vajda's Identity

Theorem
Let $F_n$ be the $n$th Fibonacci number.

Then:
 * $F_{n + i} F_{n + j} - F_n F_{n + i + j} = \left({-1}\right)^n F_i F_j$

Proof
From Fibonacci Number in terms of Smaller Fibonacci Numbers:
 * $F_{n + i} = F_n F_{i - 1} + F_{n + 1} F_i$
 * $F_{n + j} = F_n F_{j - 1} + F_{n + 1} F_j$
 * $F_{n + i + j} = F_{i - 1} F_{n + j} + F_i F_{n + j + 1}$

Therefore: