On modular representations of C-recursive integer sequences

TL;DR

This paper improves the understanding of C-recursive integer sequences by establishing a mod-mod representation valid for all positive integers, eliminating previous limitations and correction terms through an arithmetic shortcut.

Contribution

It demonstrates a mod-mod representation for all integers n ≥ 1 derived directly from the div-mod form, removing prior restrictions and correction terms.

Findings

A mod-mod representation holds for all integers n ≥ 1.

The new approach simplifies the representation by avoiding correction terms.

The result extends previous work by Prunescu and Sauras-Altuzarra.

Abstract

Prunescu and Sauras-Altuzarra showed that all C-recursive sequences of natural numbers have an arithmetic div-mod representation that can be derived from their generating function. This representation consists of computing the quotient of two exponential polynomials and taking the remainder of the result modulo a third exponential polynomial, and works for all integers $n \geq 1$. Using a different approach, Prunescu proved the existence of two other representations, one of which is the mod-mod representation, consisting of two successive remainder computations. This result has two weaknesses: (i) the representation works only ultimately, and (ii) a correction term must be added to the first exponential polynomial. We show that a mod-mod representation without inner correction term holds for all integers $n \geq 1$. This follows directly from the div-mod representation by an arithmetic short-cut from outside.