A remark on periods of periodic sequences modulo $m$ (2006.12002v1)

Abstract: 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 $F_n$.