Explicit Square Constructions for Odd-Length Geometric Progressions

Abstract

OEIS A092960 records the least square representable as one plus the sum of $n$ successive terms of a geometric progression, with value $0$ when no such square exists. For every odd $n\ge3$, we give an explicit $n$-term geometric progression with first term $2^n-3$ and common ratio $2$ whose sum plus $1$ equals $(2^n-2)^2$. This is an unconditional existence theorem for that explicit class of progressions. When applied to A092960 under the natural interpretation that such positive-integer ratio-$2$ progressions are admissible and that $0$ is used only as the nonexistence fallback, it gives the upper bound\[a(n)\le(2^n-2)^2\]at every odd index. The argument concerns existence and an upper bound, not an exact formula for the odd-index minima.