On the further properties of $\{U_n\}$

TL;DR

This paper explores additional properties of the sequence U_n, including a summation formula and congruences, expanding understanding of its mathematical structure.

Contribution

It introduces new summation formulas and congruences for the sequence U_n, enhancing theoretical knowledge of its properties.

Findings

Derived a summation formula for U_n.

Established several congruences involving U_n.

Provided insights into the sequence's algebraic and number-theoretic properties.

Abstract

Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum_{k=1}^{[n/2]} \b n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot]$ is the greatest integer function. In the paper we present a summation formula and several congruences involving $\{U_n\}$.