斐波那契数列性质

这边贺的

定义

$F_i = F_{i - 1} + F_{i - 2}(i \geq2), F_0 = 0, F_1 = 1$

结论

$1. \sum_{i = 1}^{n} F_i = F_{n + 2} - 1$

$2. \sum_{i = 1}^{n} F_i^2 = F_{n + 1}F_{n}$

$3. \sum_{i = 1}^{n} F_{2i - 1} = F_{2n}$

$4. \sum_{i = 0}^{n} F_{2i} = F_{2n + 1} - 1$

$5. F_{n} = F_{n - m}F_{m - 1} + F_{n - m + 1}F_{m}$

$6. F_{n - 1}F_{n + 1} = F_{n}^2 + (-1) ^ n$

$7. \gcd(F_{i}, F_{i - 1}) = 1$

$8. \gcd(F_{n}, F_{m}) = F_{gcd(n, m)}$

$9.$ 在 $\pmod p$ 下，斐波那契数列循环节长度 $\leq 6p$。