Progressive Cumulative Counting with Base-10 Transition and Layered Structure (Version 4)

Key Points

• The aim is to present a refined version of the Progressive Cumulative Counting sequence that transitions from odd numbers to a cumulative sum.
• Defined a deterministic two-phase process for sequence generation.
• Identified the initial phase producing consecutive odd numbers for n from 1 to 9.
• Introduced a closed-form expression for n >= 10 reflecting a shifted triangular number sequence.
• Initial phase generates odd numbers: a(n) = 2n - 1 for n <= 9.
• Cumulative transition begins at n = 10, where a(n) = a(n-1) + n applies.
• Closed-form expression yields a(n) = n(n+1)/2 - 19 for n >= 10, reflecting a quadratic growth.