Abstract
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 ⊖.
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 |
Externally published | Yes |
Keywords
- Binary sequence
- Binary tree
- Leaf depth
- Nonassociativity
- Parenthesization
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics