On the regularity of certain 1-additive sequences (Q1186116)

Given two positive integers \(u<v\), the 1-additive sequence with base \(\{u,v\}\) is the infinite sequence \((u,v)=a_ 1,a_ 2,a_ 3,\dots\) defined by \(a_ 1=u\), \(a_ 2=v\), and for \(n>2\), \(a_ n\) is the smallest integer which exceeds \(a_{n-1}\) and which has a unique representation \(a_ i+a_ j=a_ n\), \(i<j\). A 1-additive sequence is regular if successive differences \(a_{n+1}-a_ n\) are eventually periodic, i.e., there is a positive integer \(N\) such that \(a_{N+n+1}- a_{N+n}=a_{n+1}-a_ n\) for all sufficiently large \(n\). The smallest such \(N\) is the period and the value \(D=a_{N+n}-a_ n\) for large \(n\) is the fundamental difference. The major interest seems to be to determine when \((u,v)\) is regular, and then to find \(N\) and \(D\). The author surveys what is known, particularly evidence to support the conjecture that \((2,v)\) has precisely two even terms for odd \(v>3\). This is of interest, since a 1-additive sequence must be regular if it has only finitely many even terms. Exploiting a connection with the theory of homogeneous linear recurring sequences, the author uses this conjecture to obtain formulas for \(N\) and \(D\). More generally, the size of \(| N- D/4|\) is studied asymptotically.