Abstract:

In this note, we study Fibonacci-like sequences that are defined by the recurrence $S_k = a$, $S_{k + 1} = b$, $S_{n + 2} \equiv S_{n + 1} + S_n$ (mod $n + 2$) for all $n \geq k$, where $k, a, b \in \mathbb{N}$, $0 \leq a < k$, $0 \leq b < k+ 1$, and $(a,b)\ne (0,0)$. We will show that the number $\alpha = 0.S_k S_{k+1} S_{k+2} \cdots$ is irrational. We also propose a conjecture on the pattern of the sequence $\{S_n\}_{n \geq k}$.