\documentclass[12pt]{article}

\usepackage{amsmath}
\usepackage{amsfonts}

\begin{document}

\begin{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 $\{S_n\}_{n
\geq k}$.
\end{abstract}

\end{document}