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

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.