D'Ocagne's Identity
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Theorem
Let $F_k$ be the $k$th Fibonacci number.
Then:
- $\forall m, n \in \Z: \paren {-1}^n F_{m - n} = F_m F_{n + 1} - F_n F_{n - 1}$
Proof
\(\ds \paren {-1}^n F_i F_j\) | \(=\) | \(\ds F_{n + i} F_{n + j} - F_n F_{n + i + j}\) | Vajda's Identity | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-1}^n F_{m - n} F_1\) | \(=\) | \(\ds F_{n + \paren {m - n} } F_{n + 1} - F_n F_{n + \paren {m - n} + 1}\) | setting $i \gets m - n$ and $j \gets 1$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-1}^n F_{m - n}\) | \(=\) | \(\ds F_m F_{n + 1} - F_n F_{m + 1}\) | Definition of Fibonacci Numbers: $F_1 = 1$ |
$\blacksquare$
Source of Name
This entry was named for Philbert Maurice d'Ocagne.