# A note on congruence properties of the generalized bi-periodic Horadam sequence

## TL;DR

This paper explores the congruence properties of a generalized bi-periodic Horadam sequence, which varies its recurrence relation based on the parity of n, extending classical sequence analysis.

## Contribution

It introduces a generalized bi-periodic Horadam sequence with arbitrary initial conditions and analyzes its congruence properties, expanding understanding of such sequences.

## Key findings

- Identifies specific congruence relations for the sequence
- Provides formulas linking sequence terms modulo integers
- Extends classical Horadam sequence properties

## Abstract

In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1 + cw_n-2; if n is odd with arbitrary initial conditions w_0, w_1 and nonzero real numbers a, b, and c. We investigate some congruence properties of the generalized Horadam sequence {w_n}.