NOVEL SEQUENCE SPACES VIA TRIBONACCI AND LACUNARY SEQUENCES: THEORETICAL INSIGHTS, STRUCTURAL PROPERTIES, AND NEURAL NETWORK COMPARISON

The Tribonacci sequence, an extension of the Fibonacci sequence, exhibits rapid growth and a distinctive recursive structure. It plays a significant role in algebra, combinatorics, number theory, dynamical systems, and fractal geometry. Moreover, its applications extend to computer science — for modeling recursive algorithms, cryptographic structures — and to the natural sciences, such as population dynamics and biological growth patterns. In this study, we construct new sequence spaces based on the fusion of Tribonacci and lacunary sequences, generalized through modulus functions under appropriate conditions. We focus particularly on [Formula: see text]-Tribonacci lacunary statistical convergence of order [Formula: see text] and [Formula: see text]-strong Tribonacci lacunary summability of order [Formula: see text]. To strengthen our theoretical contributions, we integrate Neural Networks as computational tools for analyzing the convergence behavior and structural relationships among these sequence spaces. Several illustrative examples, supported by numerical simulations and figures, demonstrate the practical interpretation and effectiveness of the proposed concepts. Furthermore, refinements in lacunary sequences allow us to establish deeper interrelations, contributing to a more comprehensive understanding of these spaces and their applications in both mathematical theory and computational frameworks.