{"id":"palomar/PALOMAR-2026-10-17-000013","url":"https://github.com/stalex444/triangle-group-growth/tree/874b6a264e1eb8adf06141686eabab8a7afbe583","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-17-000013-v1.json","repo":"https://github.com/stalex444/triangle-group-growth","tags":["math.GR","gr-qc","math-ph","msc:20F55","msc:20F69","msc:05A15","msc:11B39","msc:83C75","msc:17B67","coxeter-groups"],"score":0,"title":"Growth series of the triangle Coxeter groups (p, q, infinity), with the PGL(2,Z) instance: Padovan sphere sizes and growth rate the plastic number","trust":"high","domain":"math.GR","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.GR","summary":"Kernel-checked proof, for every Coxeter system on three generators a, b, c with m(a,c) = p, m(b,c) = q, m(a,b) = infinity and p, q >= 2, of the growth-series identity in the ring of integer power series: (sum_n a(n) X^n) * (1 - (X + ... + X^(p-1))(X + ... + X^(q-1))) = (1 + X)(1 + ... + X^(p-1))(1 + ... + X^(q-1)), where a(n) is the number of group elements of Coxeter length n; together with its instances (2,3,inf), (2,4,inf), (3,3,inf) and, for (2,3,inf) = PGL(2,Z), the sphere sizes 1, 3, 5, 7, 9, 12, 16, ..., the Padovan identity a(n) = Pad(n+5) for n >= 2, the two-sided bounds 2 rho^n =…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-17","reactions":{},"formalized":true,"formalizes":[{"title":"Growth of planar Coxeter groups, P.V. numbers, and Salem numbers","identifier":"Mathematische Annalen 293 (1992), 475-483","relationship":"adapts"}]}