广义Fibonacci数列的平方和公式

The Squared Sum Formula for Generalized Fibonacci Number Sequence

摘要: 利用广义Fibonacci数列的递推性质, 采用初等方法证明了广义Fibonacci数列的几个平方和公式: \sum\limits_k = 1^n G_k^2 、\sum\limits_k = 1^n \left( - 1 \right)^kG_k^2 、\sum\limits_k = 1^n kG_k^2 、\sum\limits_k = 1^n G_kG_k + 1 。

Abstract: According to the definition and the recurrence property of generalized Fibonacci sequences\left\ G_n\rm \right\:G_n + 1 = uG_n + vG_n - 1, G₀=a, G₁=b, where, some quadratic sum formulas of generalized Fibonacci numbers are proved, which are\sum\limits_k = 1^n G_k^2 、\sum\limits_k = 1^n \left( - 1 \right)^kG_k^2 、\sum\limits_k = 1^n kG_k^2 、\sum\limits_k = 1^n G_kG_k + 1 .