斐波那契数列性质

定义

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

结论

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

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

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

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

\(5. F_{n} = F_{n - m}F_{m - 1} + F_{n - m + 1}F_{m}\)

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

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

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