The Challenge of Computing Geode Numbers

TL;DR

This paper explores the computational challenges of Geode numbers, a new combinatorial sequence, highlighting their complexity in higher dimensions and the difficulty of calculating large terms.

Contribution

It introduces the problem of computing Geode numbers in multiple dimensions and discusses the computational methods and challenges involved.

Findings

Closed-form expression for 2D case

Significant computational difficulty in 3D and higher

Inability to compute large terms like the 1000th in 4D

Abstract

In a fascinating recent American Mathematical Monthly article, Norman Wildberger and Dean Rubine introduced a new kind of combinatorial numbers, that they aptly named the ``Geode numbers''. While their definition is simple, these numbers are surprisingly hard to compute, in general. While the two-dimensional case has a nice closed-form expression, that make them easy to compute, already the three-dimensional case poses major computational challenges that we do meet, combining experimental mathematics and the holonomic ansatz. Alas, things get really complicated in four and higher dimensions, and we are unable to efficiently compute, for example, the $1000$-th term of the four-dimensional diagonal Geode sequence. A donation of $100$ US dollars to the OEIS, in honor of the first person to compute this number, is offered.