Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.3

The Nonassociativity of the Double Minus Operation

Jia Huang, Madison Mickey, and Jianbai Xu
Department of Mathematics and Statistics
University of Nebraska at Kearney
Kearney, NE 68849


The sequence A000975 in the Encyclopedia of Integer Sequences 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 ab := -a - b. We show that the number of distinct results obtained by inserting parentheses in the expression x0x1 ⊖ · · · ⊖ 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 .

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(Concerned with sequences A000217 A000975 A048702 A155051 A265158.)

Received May 27 2017; revised versions received August 31 2017; October 13 2017. Published in Journal of Integer Sequences, October 29 2017.

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