

A102508


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.


2



1, 3, 7, 13, 21, 31, 39, 57, 73, 91, 95, 133
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OFFSET

1,2


COMMENTS

a(n) <= n(n1)+1. Moreover, a(n)=n(n1)+1 iff A058241(n)>0, i.e., when a perfect difference set modulo n(n1)+1 exists. In particular, a(12) = 133, a(14)=183, a(17)=273, etc.
This problem is a circular analog of an optimal ruler problem; see A004137.  David Wasserman, Apr 15 2008
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


LINKS

Table of n, a(n) for n=1..12.
Don Reble, C++ Program


EXAMPLE

a(5)=21 because if we have 21 chairs, 5 persons can sit down on chairs 1, 4, 5, 10 and 12. 1=54 (mod 21). 2=1210 (mod 21). 3=41 (mod 21). 4=51 (mod 21). 5=105 (mod 21). 6=104 (mod 21). 7=125 (mod 21). 8=124 (mod 21). 9=101 (mod 21). 10=112 (mod 21). It is impossible to do the same with 22 or more chairs.


CROSSREFS

Cf. A004137, A058241.
Sequence in context: A235532 A195020 A169627 * A115298 A161206 A025728
Adjacent sequences: A102505 A102506 A102507 * A102509 A102510 A102511


KEYWORD

nonn,more,hard


AUTHOR

Ard Van Moer (ard.van.moer(AT)vub.ac.be), Mar 15 2005


EXTENSIONS

3 more terms from David Wasserman, Apr 15 2008
Edited by Max Alekseyev, Apr 29 2010, Mar 01 2015
a(11) = 95 from Don Reble, Feb 25 2015.  N. J. A. Sloane, Mar 01 2015
a(12) from Max Alekseyev, Mar 01 2015


STATUS

approved



