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A085369
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Cutting sequence for 1/e.
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2
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1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0
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OFFSET
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1,1
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COMMENTS
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Through any A085368(n) number of terms in the cutting sequence, A007677(n-1) of those terms are zeros and A007676(n) are ones. Check: A085368(5) = 26, the sequence being 3, 4, 11, 15, 26, ... (sum of numerators and denominators of convergents to 1/e). Then through n=26, A085369(n) is 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0, with 7 zeros and 19 ones, (7/19 being the 5th convergent to 1/e): 7/19 = [2, 1, 2, 1, 1]. Numerator and denominator sum = 26, with 7 zeros and 19 ones, with the zeros occupying positions n = 3, 7, 11, 14, 18, 22 and 26 (also being the first 7 terms of A000572). Positions of the cutting sequence occupied by ones (1, 2, 4, 5, 6, ...) are consecutive terms of the lower Beatty sequence A006594, being generated by floor(n*(1 + 1/e)).
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REFERENCES
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Manfred R. Schroeder, "Fractals, Chaos, Power Laws", Freeman, 1996, p. 56.
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LINKS
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FORMULA
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Given the line y = (1/e)x starting from (0, 0) and passing through an array of squares, a "1" denotes an intersection with a vertical line, while an "0" denotes an intersection with a horizontal line.
n for 0's are consecutive terms of upper Beatty pair terms A000572: 3, 7, 11, 14, 18, 22, 26, ..., while n's for all 1's are paired lower Beatty terms of A006594: 1, 2, 4, 5, 6, 8, ...
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EXAMPLE
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a(6) = 1, where 1's correspond to members of the lower Beatty pair A006594 which is generated from floor(n*(1 + 1/e)). Check: floor(5*(1 + 1/e)) = 6. All terms not in A006594 are 0's.
a(7) = 0, where 7 is not a member of A006594, but is a member of the upper Beatty pair sequence A000572 which has the generator floor(n*(e + 1)). Check: floor(2*(1 + e)) = 7.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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