# Zenodo Upload — Metadata ## Title From the Quadratic Sequence to the Conical Helix ## Creators Name: Balban, Dogan Affiliation: Independent Researcher ORCID: 0009-0002-5052-6951 ## Description This work develops a complete analytical and geometric framework for the quadratic integer sequence k(n) = (2n² + 3n + 1) / 6, which takes integer values precisely when n ≡ 1, 5 (mod 6). A fine-grained analysis of the first differences reveals an alternating pattern of minor and major steps whose pairwise sums form — as proved in this work — the unique partition of the difference sequence into equal-length blocks that yields an arithmetic progression of circle circumferences. This algebraic uniqueness forces a constant radial increment ΔR = 12/π and leads, without any further degree of freedom, to an explicit Archimedean spiral with slope a = 6/π² = 1/ζ(2). Extending the planar spiral by a linear height function produces a three-dimensional conical helix on a right circular cone with half-angle α = arctan(2/π) ≈ 32.48°. The algebraic core of the entire construction is the fundamental identity (4n+3)² = 48 k(n) + 1, which defines a universal quantity Q(k) = √(48k+1) from which radius, azimuth, and height of the helix can be expressed as exact closed functions of a single parameter. As a corollary of the congruence-class structure, all primes p > 3 appear as a distinguished subset of the integer-indexed points on the helix. The repository contains the full LaTeX source code (20 chapters), all figures, Python verification scripts, and the compiled PDF. ## Keywords quadratic integer sequence Archimedean spiral conical helix congruence classes prime numbers parametric geometry arc-length parametrisation Basel problem zeta(2) ## Language eng (Main text English) ## License Creative Commons Attribution 4.0 International (CC BY 4.0) ## Upload Type Publication ## Publication Date 2026-02-28 ## Related Identifiers (GitHub Repository) https://github.com/dogan1908/quadratichelix_en ## MSC 2020 11B25 — Arithmetic progressions 53A04 — Curves in Euclidean and related spaces 11A41 — Primes 11B83 — Special sequences and polynomials ## Suggested Citation Balban, D. (2026). From the Quadratic Sequence to the Conical Helix. Zenodo. https://doi.org/10.5281/zenodo.18905795 ## BibTeX @misc{Balban2026, author = {Balban, Dogan}, title = {From the Quadratic Sequence to the Conical Helix}, year = {2026}, publisher = {Zenodo}, doi = {10.5281/zenodo.18905795}, url = {https://doi.org/10.5281/zenodo.18905795} }