What question did this study set out to answer?

The research aims to analyze a family of linear recurrent sequences with negative coefficients and their limiting behavior.

March 22, 2026Open Access

A Family of Recurrent Sequences with Oscillatory Stabilization and the √ 2 Limit

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Key Points

The research aims to analyze a family of linear recurrent sequences with negative coefficients and their limiting behavior.
Studied linear recurrent sequences defined by a single negative coefficient at the first preceding term.
Derived analytical properties related to characteristic polynomials.
Conducted high-precision numerical calculations to support findings.
The dominant roots' moduli of these sequences converge to √2 as the system order increases.
This behavior contrasts with classical results like Pisot numbers, which converge to 2.

Abstract

Abstract This paper investigates a specific family of linear recurrent sequences characterized by a single negative coefficient at the first preceding term. It is shown that the characteristic polynomials of such systems possess a unique property: as the order of the system increases, the moduli of their dominant roots monotonically converge to √2 This result contrasts with classical cases, such as Pisot numbers, where the limit is 2. The findings are supported by high-precision numerical calculations and an analytical derivation.

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Emma Helmdach

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A Family of Recurrent Sequences with Oscillatory Stabilization and the √ 2 Limit

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