On modular representations of C-recursive integer sequences

Mihai Prunescu; Joseph M. Shunia

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arXiv · 2502.16928

On modular representations of C-recursive integer sequences

Mihai Prunescu·Joseph M. Shunia

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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.

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Mihai Prunescu, Joseph M. Shunia. 2025-02-24. On modular representations of C-recursive integer sequences. https://arxiv.org/abs/2502.16928

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