The Nonassociativity of the Double Minus Operation

The sequence A000975 in OEIS can be defined by A1 = 1, An+1 = 2An if n is odd, and An+1 = 2An + 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⊖ b := −a− b. We show that the number of distinct results obtained by inserting parentheses in the expression x0 ⊖ x1 ⊖ · · · ⊖ xn equals An, by investigating the leaf depth in binary trees. Our result can be viewed as a quantitative measurement for the nonassociativity of the binary operation ⊖.