Quadratic-periodic formula for general three-term Gale–Robinson growth sequences

Prove that, for every three-term Gale–Robinson growth rate sequence defined by integers 1≤p<q<r with m=p+q+r and gcd(p,q,r)=1, the identity k_n=(n^2+c_n)/(2N) holds for all n≥1, where c_n is periodic with period N=gcd(p,q)r^2+gcd(p,r)q^2+gcd(q,r)p^2+pqr.

Background

For a three-term Gale–Robinson recurrence, the paper associates a max-linear growth rate sequence with initial values k_1=…=k_{m-1}=0 and k_m=1. The conjecture predicts that every such sequence with coprime parameters is quadratic up to a periodic correction.

The formula would provide an explicit degree-growth law and, through the paper’s entropy formula, refine the known result that the associated rational maps have zero algebraic entropy. The paper proves the conjecture only for the infinite subfamily p=1, q=2, and r≥3; the general coprime-parameter case is therefore left unresolved.