Written: Aug. 13, 2025

[To appear in Palestine Journal of Mathematics]

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.

Maple package

• Geode.txt, A Maple package to compute, very, efficiently, 3-D Geode numbers, but much less efficiently, higher dimensional ones.

Sample input and output for Geode.txt (and some SAGE output)

• If you want to see the first 1000 terms of the 3D diagonal Geode sequence, i.e. G([i,i,i]) for i from 1 to 1000

  input file leads to the   output file

• If you want to see The Geode numbers for [10^i,10^i,10^i], for i from 1 to 5,

  input file leads to the   output file

  Here is G([10^6,10^6,10^6]), using Sage.

• If you want to see The Geode numbers for [100*i,200*i,300*i], for i from 1 to 3

  input file leads to the   output file

• If you want to see the first 30 terms of the 4-dimensional Geode sequence

  input file leads to the   output file

  Here are 100 terms (computed with SAGE)

• If you want to see the first 20 terms of the 5-dimensional diagonal Geode sequence

  input file leads to the   output file

  Here are 29 terms (computed with SAGE)

• If you want to see the first 10 terms of the 6-dimensional diagonal Geode sequence

  input file leads to the   output file

• If you want to see the 2nd-order recurrences in the format [[f1,f2],[a1,a2]] (encoding the unique sequence a(n) such that a(n)=f1(n)*a(n-1)+f2(n)*a(n-2), with initial conditions a(1)=a1, a(2)=a2), for the 3D Geode numbers {G([n+c1,n+c2,n+c3]), n=1..infinity} where the list of length 7, Rec3, is such that i-1=c1*4+c2*2+c3

  input file leads to the   output file

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