Between Consecutive Squares: The Arithmetic of A392243

Between Consecutive Squares develops the arithmetic of OEIS sequence A392243, defined by the signed sum a(n) = sum_{i=1}^n i(-1)^{ceil(sqrt(i))}, whose signs change immediately after each perfect square. The book begins from an elementary closed formula for the sequence and follows its consequences through a connected series of questions: why cubes occur at square indices, why triangular magnitudes appear at the central crossing, how to find every occurrence of a given value, how perfect squares and sixth powers arise, and why Fibonacci numbers produce an infinite family of nonforced sixth-power values. Later chapters study the distribution and multiplicity of attained magnitudes, repetitions between different square intervals, a recurrence that reproduces the absolute values of A392243, weighted analogues, and the cube-path geometry obtained by moving the sign-change positions. Pell equations, divisor factorizations, asymptotic counting, polynomial reflection identities, and selected computer-assisted elliptic-curve calculations appear naturally along the way. The exposition is intended to remain accessible from ordinary algebra. More technical collision arguments, weighted extensions, and analytic refinements are placed in appendices so that they can be deferred on a first reading. Proved results, finite computations, computer-assisted classifications, conjectures, and open questions are kept explicitly separate. This release contains the final 80-page reviewed reading edition, a complete AI-readable Markdown version of the book, the canonical manuscript source, and a versioned computational companion containing preserved scripts, outputs, environment information, and reproducibility records for the computer-assisted results. The research was developed with substantial assistance from AI systems, including Claude, Astra, and GPT-5.6 Sol. The author is responsible for the mathematical presentation and final claims. AI-assisted internal review is documented as such and is not represented as external peer review.