Vajda's Identity

Theorem
Let $F_k$ be the $k$th Fibonacci number.

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

Proof
From Fibonacci Numbers 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: