OEIS A000109: Planar Triangulations, Simplicial Polyhedra, and the Geometry–Graph–Number Theory Bridge

We dive into OEIS A000109, the count of simplicial polyhedra (triangular-faced 3D shapes) with n vertices. These numbers are in bijection with maximal planar graphs on n vertices, i.e., simple planar graphs that are triangulated to the max. Through duality, planar triangulations connect to 3-connected cubic planar graphs, tying geometry, graph theory, and number theory together. We discuss lower bounds, asymptotic growth, and why there is no simple closed form. PlanTree, the specialized program, is introduced as a key tool to generate these graphs and extend the sequence. We also explore the rich web of applications and connections to computer graphics, chemistry (fullerenes), architecture, and physics, illustrating how a single integer sequence weaves together seemingly distant areas of mathematics.

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