Three-Quarter Square Numbers: Algebraic Structures, Combinatorial Properties, and Geometric Applications

Abstract

Figurate numbers provide important connections between algebraic structures, geometric configurations, and combinatorial patterns. This study presents a systematic investigation of Three-Quarter Square Numbers (OEIS A077043), a fractional figurate sequence defined by . Using a descriptive-expository mathematical approach, the study examines the algebraic properties, figurate relationships, and combinatorial and geometric interpretations of the sequence. The investigation establishes the parity-based closed form, difference patterns, recurrence relations, and generating function representations that reveal the underlying regularity of the sequence. Furthermore, the study identifies significant connections between Three-Quarter Square Numbers and classical figurate sequences, including triangular, centered triangular, centered hexagonal, pentagonal, pronic, star, and octagonal numbers. The sequence also shown to arise naturally in several discrete mathematical contexts, including maximum triple intersections of three-line families, enumeration of noncongruent integer-sided isosceles triangles, maximal coefficients of polynomial expansions of the form , and partitions of  into exactly three positive parts. The findings establish the Three-Quarter Square Numbers as a structurally rich fractional figurate sequence that unifies algebraic, combinatorial, and geometric settings, contributing to a broader understanding of generalized figurate number theory.