### 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) |
---|---|

Article number | 17.10.3 |

Journal | Journal of Integer Sequences |

Volume | 20 |

Issue number | 10 |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Integer Sequences*,

*20*(10), [17.10.3].

**The nonassociativity of the double minus operation.** / Huang, Jia; Mickey, Madison; Xu, Jianbai.

Research output: Contribution to journal › Article

*Journal of Integer Sequences*, vol. 20, no. 10, 17.10.3.

}

TY - JOUR

T1 - The nonassociativity of the double minus operation

AU - Huang, Jia

AU - Mickey, Madison

AU - Xu, Jianbai

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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 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 ⊖.

AB - 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 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 ⊖.

KW - Binary sequence

KW - Binary tree

KW - Leaf depth

KW - Nonassociativity

KW - Parenthesization

UR - http://www.scopus.com/inward/record.url?scp=85034214669&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85034214669&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85034214669

VL - 20

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 10

M1 - 17.10.3

ER -