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A269970
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Factorial-nested interval sequence of 1/e.
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12
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2, 1, 2, 3, 1, 2, 2, 1, 3, 2, 1, 2, 4, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 3, 2, 4, 1, 2, 2, 1, 2, 2, 3, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 3
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OFFSET
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1,1
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COMMENTS
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Suppose that r = (r(n)) is a sequence satisfying (i) 1 = r(1) > r(2) > r(3) > ... and (ii) r(n) -> 0. For x in (0,1], let n(1) be the index n such that r(n+1) , x <= r(n), and let L(1) = r(n(1))-r(n(1)+1). Let n(2) be the index n such that r(n(1)+1) < x <= r(n(1)+1) + L(1)r(n), and let L(2) = (r(n(2))-r(r(n)+1)L(1).
Continue inductively to obtain the sequence (n(1), n(2), n(3), ... ), the r-nested interval sequence of x. Taking r = (1/n!) gives the factorial-nested interval sequence of x.
Conversely, given a sequence s= (n(1),n(2),n(3),...) of positive integers, the number x having satisfying NI(x) = s is the sum of left-endpoints of nested intervals (r(n(k)+1), r(n(k))]; i.e., x = sum{L(k)r(n(k+1)+1), k >=1}, where L(0) = 1.
Guide to related sequences:
x factorial-nested interval sequence
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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