Five mutually tangent spheres, three unit balls between two plates, repeatedly reflected into the gaps. Mallows counted the spheres added in each generation two ways. Strategy (a) follows reflection words; strategy (b) also uses the extra Descartes quintuples that close up early. Drag to orbit, scroll to zoom, hover a sphere for its bend.
Each sphere is an exact integer vector, so nothing here depends on floating point, and every corrected term was recomputed a second time by plain geometry. Struck terms disagree with the recomputation; green terms replace them; gray terms are new. Proofs, checks and programs: oeis-results/A154640.
| Entry | Counts | Terms from n = 0 | Result |
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Descartes reflections act on the integer sphere vectors exactly as the Coxeter group with five generators and all mij = 3 acts in its Tits representation. Each starting sphere is a vertex of the fundamental chamber, fixed by the four other reflections. By Tits's theorem every sphere is reached from exactly one starting sphere j and one coset of that four-generator subgroup. The generation of a sphere is the length of the shortest element in its coset, so the counts are the growth series of the group divided by the subgroup's, times five.
In the second generation, ten pairs of new spheres touch each other, closing up ten extra quintuples (three old spheres and two new). Strategy (b) reflects these at once, so their spheres arrive a generation early, and the effect compounds. Mallows's own rule for this (a word of length k with j disjoint "aba" factors belongs to generation k − j) agrees with the geometry for every element checked; his generation-5 values were a slip in carrying it out.