A Classification of Motzkin Numbers Modulo 8

Abstract

The well-known Motzkin numbers were conjectured by Deutsch and Sagan to be nonzero when modulo $8$. The conjecture was first proved by Sen-Peng Eu, Shu-chung Liu and Yeong-Nan Yeh by using the factorial representation of the Catalan numbers. We present a short proof by finding a recursive formula for Motzkin numbers modulo $8$. Moreover, such a recursion leads to a full classification of Motzkin numbers modulo $8$.

An addendum was added on April 3 2018.