Consider the recurrence equation defined by and
(1)
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where is the floor function. Graham and Pollak actually defined , but the indexing will be used here for convenience, following Borwein and Bailey (2003, p. 62). The first few terms are summarized in the following table for small values of .
OEIS | , , ... | |
1 | A001521 | 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 54, ... |
5 | A091522 | 5, 7, 10, 14, 20, 28, 40, 57, 81, 115, ... |
8 | A091523 | 8, 12, 17, 24, 34, 48, 68, 96, 136, 193, ... |
Amazingly, an explicit formula for with is given by
(2)
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where is the th smallest number in the set (Graham and Pollak 1970; Borwein and Bailey 2003, p. 63).
Now consider the associated sequence
(3)
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whose value is always 0 or 1. Even more amazingly, interpreting the sequence as a series of binary bits gives a series of algebraic constants
(4)
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where the first few constants are
(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(OEIS A091524 and A091525; Borwein and Bailey 2003, p. 63).
It is not known if sequences such as
(15)
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(16)
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have corresponding properties (Graham and Pollak 1970; Borwein and Bailey 2003, p. 63).