A combinatorial proof of a recursive relation of the Motzkin sequence by lattice paths (Q2785021)

Let \(m_n\) be the \(n\)th Motzkin number. The first few Motzkin numbers are \(1,1,2,4,9,21,51,\dots\) In this paper the author gives a combinatorial proof of the recursion of the Motzkin sequence showing that \((n+ 4)m_{n+2}= (2n+5)m_{n+1}+ 3(n+1)m_n\) \((n\geq 0)\), and also that \(\lim_{n\to\infty} m_{n+1}/m_n= 3\).