This document presents Version 5 (V5) of the Progressive Cumulative Counting framework, extending the construction to an entire family of sequences parameterized by the base B. For each base B ≥ 2, the sequence aB (n) is defined by a two-phase process: an initial linear phase generating consecutive odd numbers, followed by a structural transition at n = B - 1, after which the sequence evolves cumulatively as a (n) = a (n-1) + n. For n ≥ B - 1, the sequence admits a closed-form expression: aB (n) = n (n+1) /2 + K (B), where K (B) = (-B² + 7B - 8) /2. A structural symmetry emerges: if two bases B1 and B2 satisfy B1 + B2 = 7, then the corresponding sequences coincide for all n ≥ max (B1 - 1, B2 - 1). This establishes a direct connection between base-dependent constructions and shifted triangular number sequences, revealing a simple but nontrivial algebraic structure.
Andrea Esposito (Sun,) studied this question.