Tournament sequences and Meeussen sequences (Q1583614)

A tournament sequence [see \textit{P. Capell} and \textit{T. V. Narayana}, Can. Math. Bull. 13, 105-109 (1970; Zbl 0225.60006)] is an increasing sequence of positive integers \((t_1, t_2,\dots)\) such that \(t_1+1\) and \(t_{i+1}\leq 2t_i\). The authors define a Meeussen sequence to be an increasing sequence of positive integers \((m_1, m_2,\dots)\) such that \(m_1=1\), every non-negative integer is the sum of a subset of the \(\{m_i\}\) and each integer \(m_i-1\) is the sum of a unique such subset. They then show that Meeussen sequences are precisely the tournament sequences, by exhibiting a bijection between the two sets of sequences which respects the natural tree structure on each set. They also present an efficient way of counting these sequences, and discuss the asymptotic growth of the number of sequences.