A New Family of Somos-like Recurrences

TL;DR

This paper introduces a new family of quadratic recurrence relations inspired by Somos sequences, proving integrality for a subfamily and conjecturing parameter relations for integer sequences.

Contribution

It presents a three-parameter family of Somos-like recurrences, proves integrality for a specific subfamily via linear recurrence equivalence, and conjectures conditions for integer sequences.

Findings

Proposed a three-parameter family of quadratic recurrences.

Proved integrality for a subfamily using linear recurrence equivalence.

Conjectured parameter relations for generating integer sequences.

Abstract

We exhibit a three parameter infinite family of quadratic recurrence relations inspired by the well known Somos sequences. For one infinite subfamily we prove that the recurrence generates an infinite sequence of integers by showing that the same sequence is generated by a linear recurrence (with suitable initial conditions). We also give conjectured relations among the three parameters so that the quadratic recurrences generate sequences of integers.