On the representation of C-recursive integer sequences by arithmetic terms

TL;DR

This paper demonstrates that integer sequences defined by linear recurrences with rational coefficients can be expressed as differences of two exponential arithmetic terms without irrational constants, with applications to Lucas, Fibonacci, and Pell sequences.

Contribution

It introduces a novel representation of C-recursive sequences as differences of two exponential arithmetic terms, expanding understanding of their structure.

Findings

Sequences can be represented as differences of two exponential arithmetic terms.

Applicable to Lucas, Fibonacci, and Pell sequences.

No irrational constants are involved in the representation.

Abstract

We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant. We apply our methods to various Lucas sequences including the classical Fibonacci sequence, to the sequence of solutions of the Pell equation and to some natural C-recursive sequences of degree 3.