The Nonassociativity of the Double Minus Operation

Abstract

The sequence A000975 in OEIS can be defined by $A_1=1$, $A_{n+1}=2A_n$ if $n$ is odd, and $A_{n+1}=2A_n+1$ if $n$ is even. This sequence satisfies other recurrence relations, admits some closed formulas, and is known to enumerate several interesting families of objects. We provide a new interpretation of this sequence using a binary operation defined by $a\ominus b:=-a-b$. We show that the number of distinct results obtained by inserting parentheses in the expression $x_0\ominus x_1\ominus \cdots \ominus x_n$ equals $A_n$, by investigating the leaf depth in binary trees. Our result can be viewed as a quantitative measurement for the nonassociativity of the binary operation $\ominus$.