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A102508
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Suppose there are equally spaced chairs around a round table. Then a(n) is the maximal number of chairs for which there exists a seating arrangement of n people around the table such that if a waiter puts two glasses (randomly) on the table in front of two (different) chairs, it is always possible to turn the table so that the two glasses end up in front of two seated persons.
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2
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1, 3, 7, 13, 21, 31, 39, 57, 73, 91, 95, 133
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OFFSET
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1,2
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COMMENTS
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a(n) <= n(n-1)+1. Moreover, a(n)=n(n-1)+1 iff A058241(n)>0, i.e., when a perfect difference set modulo n(n-1)+1 exists. In particular, a(12) = 133, a(14)=183, a(17)=273, etc.
Solutions do not always exist for table sizes less than a(n). For example, for n = 5 there is no solution for a table of size 20. - David Wasserman, Apr 15 2008
Equivalently, largest value of S such that in some cyclic array of positive integers of length n, every positive integer <= S is the sum of consecutive terms. For example, the numbers 1..21 can be written as the sum of consecutive terms in the cyclic array [10,3,1,5,2]. So a(5) = 21. - Phil Scovis, Jan 29 2016
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LINKS
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EXAMPLE
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a(5)=21 because if we have 21 chairs, 5 persons can sit down on chairs 1, 4, 5, 10 and 12. 1 == 5-4 (mod 21). 2 == 12-10 (mod 21). 3 == 4-1 (mod 21). 4 == 5-1 (mod 21). 5 == 10-5 (mod 21). 6 == 10-4 (mod 21). 7 == 12-5 (mod 21). 8 == 12-4 (mod 21). 9 == 10-1 (mod 21). 10 == 1-12 (mod 21). It is impossible to do the same with 22 or more chairs.
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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Ard Van Moer (ard.van.moer(AT)vub.ac.be), Mar 15 2005
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EXTENSIONS
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STATUS
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approved
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