### A Generalization of Very Odd Sequences

#### Abstract

Let $\mathbb N$ be the set of positive integers and $n\in \mathbb N$. Let $\mathbf{a}=(a_0,a_1,\dots, a_{n-1})$ be a sequence of length $n$, with $a_i\in \{0,1\}$. For $0\leq k\leq n-1$, let \[ A_k(\mathbf{a})=\sum_{\substack{0\leq i\leq j\leq n-1\\ j-i=k}} a_ia_j.\] The sequence $\mathbf{a}$ is called a very odd sequence if $A_k(\mathbf{a})$ is odd for all $0\leq k\leq n-1$. In this paper, we study a generalization of very odd sequences and give a characterisation of these sequences.

#### Keywords

very odd sequence; Pelikan’s conjecture