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Search: id:A090569
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| A090569 |
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The survivor w(n,2) in a modified Josephus problem, with a step of 2. |
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+0
3
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| 1, 1, 3, 3, 1, 1, 3, 3, 9, 9, 11, 11, 9, 9, 11, 11, 1, 1, 3, 3, 1, 1, 3, 3, 9, 9, 11, 11, 9, 9, 11, 11, 33, 33, 35, 35, 33, 33, 35, 35, 41, 41, 43, 43, 41, 41, 43, 43, 33, 33, 35, 35, 33, 33, 35, 35, 41, 41, 43, 43, 41, 41, 43, 43, 1, 1, 3, 3, 1, 1, 3, 3, 9, 9, 11, 11, 9, 9, 11, 11, 1, 1
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Arrange n persons {1,2,...,n} consecutively on a line rather than around in a circle. Beginning at the left end of the line, we remove every qth person until we reach the end of the line. At this point we immediately reverse directions, taking care not to "double count" the person at the end of the line, and continue to eliminate every qth person, but now moving right to left. We continue removing people in this back and forth manner until there remains a lone survivor w(n,q).
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REFERENCES
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Chris Groer, The Mathematics of Survival: From Antiquity to the Playground, Am. Math. Monthly 110(2003)812-825.
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FORMULA
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w(n, 2)=1+Sum[b(j)*(2^j), j=1...k, j odd], where Sum[b(j)*(2^j), j=0...k] is the binary expansion of N, with N either n or n-1, whichever is odd.
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EXAMPLE
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W(12,2)=11, since people are eliminated in the order 2, 4, 6, 8, 10, 12, 9, 5, 1, 7, 3, leaving 11 as the survivor.
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CROSSREFS
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Cf. A088443, A088452.
A063695(n-1) + 1.
Sequence in context: A096433 A084101 A053386 this_sequence A109439 A133333 A133332
Adjacent sequences: A090566 A090567 A090568 this_sequence A090570 A090571 A090572
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KEYWORD
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nonn
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AUTHOR
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John W. Layman (layman(AT)math.vt.edu), Dec 02 2003
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