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
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 =…
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