论文标题
关于周期序列时期modulo $ m $的评论
A remark on periods of periodic sequences modulo $m$
论文作者
论文摘要
令$ \ {g_n \} $为整数模型$ m $的定期顺序，让$ \ {sg_n \} $为$ sg_n：= \ sum_ {k = 0}^ng_k $（mod $ m $）定义的部分总和序列。我们给出了$ \ {sg_n \} $的公式。我们还表明，对于一般的fibonacci序列$ f（a，b）_n $，以至于$ f（a，b）_0 = a $ and $ f（a，b）_1 = b $，我们有$ s^i f（a，a，b）_n = s^{s^{i-1} f（a，b） \选择i-1} b $$其中$ s^i f（a，b）_n $是由$ s^i f（a，b）_n：= \ sum_ {k = 0}^n s^n s^{i-1} f（i-1} f（a，a，b）f（a，b）f（a，b）_k $。这是众所周知的公式$$ \ sum_ {k = 0}^n f_k = f_k = f_ {n+2} -1 $$的广义版本 fibonacci序列$ f_n $。
Let $\{G_n\}$ be a periodic sequence of integers modulo $m$ and let $\{SG_n\}$ be the partial sum sequence defined by $SG_n:= \sum_{k=0}^nG_k $ (mod $m$). We give a formula for the period of $\{SG_n\}$. We also show that for a generalized Fibonacci sequence $F(a,b)_n$ such that $F(a,b)_0=a$ and $F(a,b)_1=b$, we have $$S^i F(a,b)_n = S^{i-1}F(a,b)_{n+2}-{n+i \choose i-2}a-{n+i \choose i-1} b$$ where $S^i F(a,b)_n $ is the i-th partial sum sequence successively defined by $S^i F(a,b)_n := \sum_{k=0}^n S^{i-1}F(a,b)_k$. This is a generalized version of the well-known formula $$\sum_{k=0}^n F_k = F_{n+2} -1$$ of the Fibonacci sequence $F_n$.
