This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A102508 #45 Aug 07 2025 00:30:02 %S A102508 1,3,7,13,21,31,39,57,73,91,95,133 %N 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. %C A102508 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. %C A102508 This problem is a circular analog of an optimal ruler problem; see A004137. - _David Wasserman_, Apr 15 2008 %C A102508 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 %C A102508 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 %C A102508 If there exists a ruler of length L and n marks, then it can be trivially transformed to a ruler of length L and n+1 marks, by simply dividing one of the segments into two. In other words, a(n+1) >= a(n). - _Dmitry Kamenetsky_, Aug 02 2025 %C A102508 a(14)=183, a(17)=273, a(18)=307, a(20)=381, a(24)=553 and so on. See Dan Gordon's site in links. - _Dmitry Kamenetsky_, Aug 02 2025 %C A102508 a(16) >= 195, since every length from 1 to 195 can be generated with the cyclic ruler [3,14,2,5,29,1,4,66,6,9,11,1,1,10,8,25]. - _Dmitry Kamenetsky_, Aug 02 2025 %H A102508 Dan Gordon, <a href="https://www.dmgordon.org/diffset/">Difference Sets</a> %H A102508 Don Reble, <a href="/A102508/a102508.txt">C++ Program</a> %e A102508 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. %Y A102508 Cf. A004137, A058241. %K A102508 nonn,more,hard %O A102508 1,2 %A A102508 Ard Van Moer (ard.van.moer(AT)vub.ac.be), Mar 15 2005 %E A102508 3 more terms from _David Wasserman_, Apr 15 2008 %E A102508 Edited by _Max Alekseyev_, Apr 29 2010, Mar 01 2015 %E A102508 a(11) = 95 from _Don Reble_, Feb 25 2015. - _N. J. A. Sloane_, Mar 01 2015 %E A102508 a(12) from _Max Alekseyev_, Mar 01 2015