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

Department of Mathematics and Statistics

University of Nebraska at Kearney

Kearney, NE 68849

USA

**Abstract:**

The sequence
A000975 in the
*Encyclopedia of Integer Sequences* can be
defined by
*A*_{1} = 1,
*A*_{n+1} = 2*A*_{n}
if *n* is odd, and
*A*_{n+1} = 2*A*_{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* ⊖ *b* := -*a* - *b*. We show that the number of
distinct results obtained by inserting parentheses in the expression
*x*_{0} ⊖ *x*_{1} ⊖ · ·
· ⊖ *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 .

(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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