ALGEBRAIC STRUCTURES IN GENERALIZED FIBONACCI SEQUENCES OVER FINITE COMMUTATIVE RINGS

The Fibonacci recurrence, ordinarily studied over the integers, acquires a considerably richer algebraic character once it is reduced over a finite commutative ring. This paper examines periodicity, rank of apparition, and structural behaviour of generalized Fibonacci-type sequences defined over Z_n across a spread of moduli that include primes, prime powers, and composite integers with mixed factorisations. Building on the classical notion of the Pisano period, the analysis is extended to a family of second-order linear recurrences G(k) = c1 G(k-1) + c2 G(k-2) mod n, parametrised by coefficients c1 and c2, to examine how variation in these coefficients reshapes the period structure, particularly when c2 fails to be a unit of Z_n. Computational data gathered across thirty moduli show that the period of the standard Fibonacci recurrence is invariant under choice of initial seed whenever the seed vector is non-degenerate, a fact traceable to the companion-matrix formulation of the recurrence over the ring. Prime-power lifting exhibits a broadly multiplicative pattern, period(p^(k+1)) equal top times period(p^k) in the cases observed, subject to exceptions associated with WallSun-Sun-type behaviour. A Jacobsthal-type variant, in which the ring's zero divisors interact directly with the recurrence coefficients, is shown to become only eventually periodic rather than purely periodic, a distinction with no counterpart over fields. These findings position the generalized Fibonacci sequence as a diagnostic device for probing unit structure, zero-divisor behaviour, and matrix invertibility within finite commutative rings, with downstream relevance for pseudo-random sequence design and coding-theoretic constructions.