Recursive Block Addition of Integers and a General Index

Starting with the integers a₁ = 1 and a₂ = 2, we successively append the blocks (aₖ + a₁, …, aₖ + aₖ₋₁) for k ≥ 2, retaining every sum in the order in which it is formed. We study the resulting numerical sequence through its positions, repeated values, sums, and growth. The main result is a general index recurrence: for n ≥ 3, the two parent indices of aₙ are given explicitly by inversion of triangular block boundaries. This determines every term from smaller indices. We derive identities for partial sums, sums of squares, and prefix polynomials, together with a recurrence for the last occurrence of each positive integer. The sequence tends to infinity despite infinitely many decreases. Its lower growth scale is doubly logarithmic, while its prefix maximum has logarithmic order. Independent implementations agree through one million terms.