### Abstract

The sequence A000975 in the Encyclopedia of Integer Sequences 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 ⊖ 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 ⊖.

Original language | English (US) |
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Article number | 17.10.3 |

Journal | Journal of Integer Sequences |

Volume | 20 |

Issue number | 10 |

State | Published - Jan 1 2017 |

### Keywords

- Binary sequence
- Binary tree
- Leaf depth
- Nonassociativity
- Parenthesization

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

*Journal of Integer Sequences*,

*20*(10), [17.10.3].