Identities and congruences for a new sequence

TL;DR

This paper introduces a new integer sequence defined by a recursive relation, explores its properties, and establishes various recursive formulas and congruences involving the sequence.

Contribution

The paper presents a novel sequence with its recursive formulas and congruences, expanding the understanding of such integer sequences.

Findings

Derived multiple recursive formulas for the sequence.

Established several congruences related to the sequence.

Provided new insights into the properties of the sequence.

Abstract

Let $[x]$ be the greatest integer not exceeding $x$. In the paper we introduce the sequence $\{U_n\}$ given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]}\binom n{2k}U_{n-2k}\quad(n\ge 1)$, and establish many recursive formulas and congruences involving $\{U_n\}$.