On Octagonal Pyramidal Number

Abstract

This study investigates octagonal pyramidal numbers, a three-dimensional figurate sequence obtained by summing successive octagonal numbers. Using descriptive and expository methods, it establishes the general formula  and examines its equivalent algebraic and combinatorial forms. The study also derives ordinary and exponential generating functions, recurrence relations, summation identities, reciprocal sums, divisibility patterns, and parity properties. Furthermore, relationships between octagonal pyramidal numbers and other integer sequences, including triangular, cubic, odd, pentagonal, pentagonal pyramidal, tetrahedral, and heptagonal numbers, are presented and proven. The findings demonstrate that octagonal pyramidal numbers possess significant algebraic, combinatorial, and number-theoretic structures. These results provide a systematic foundation for further investigations involving generalized figurate numbers, computational properties, and possible applications in mathematics.