A proof of a conjecture of Knuth

Abstract

From numerical experiments, D. E. Knuth conjectured that $0<D_{n+4}<D_n$ for a combinatorial sequence $(D_n)$ defined as the difference $D_n = R_n-L_n$ of two definite hypergeometric sums. The conjecture implies an identity of type $L_n= \lfloor R_n \rfloor$, involving the floor function. We prove Knuth's conjecture by applying Zeilberger's algorithm as well as classical hypergeometric machinery.