cp's OEIS Frontend

A058241 Number of sets of n positive integers that can be placed along a circle such that the set of sums of adjacent integers forms { 1, 2, ..., n^2-n+1 }.

%I A058241 #46 Feb 16 2025 08:32:43
%S A058241 1,1,1,2,1,5,0,6,4,6,0,18,0,20,0,0,6,51,0,42,0,0,0
%N A058241 Number of sets of n positive integers that can be placed along a circle such that the set of sums of adjacent integers forms { 1, 2, ..., n^2-n+1 }.
%C A058241 a(1)=1, a(2)=1.
%C A058241 Conjecture: for n > 2, p prime, e > 0, if n-1 is of the form p^e then a(n) > 0, otherwise a(n)=0.
%C A058241 From _Zhao Hui Du_, Mar 18 2019: (Start)
%C A058241 Conjecture: for n > 2, p prime, e > 0, if n-1 is of the form p^e then a(n) = A000010(n^2-n+1)/(6e), otherwise a(n)=0.
%C A058241 If a(n) is nonzero, a finite projective plane of order n-1 could be constructed.
%C A058241 Brute-force enumeration shows a(21)=0.
%C A058241 The Bruck-Ryser Theorem shows that if a finite projective plane of order q exists and q is congruent to 1 or 2 (mod 4), then q must be the sum of two squares. We could get a(22)=a(23)=0 from that theorem.
%C A058241 (End)
%H A058241 Leonard E. Dickson, <a href="https://doi.org/10.2307/2968498">Problem 142</a>, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), pp. 107-108.
%H A058241 D. Mehendale, <a href="https://arxiv.org/abs/math/0611492">Finite Projective Planes</a>, arXiv:math/0611492 [math.GM], 2006-2015.
%H A058241 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectDifferenceSet.html">Perfect Difference Set</a>
%H A058241 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bruck%E2%80%93Ryser%E2%80%93Chowla_theorem">Bruck-Ryser theorem</a>
%e A058241 For n=3, we can choose a set { 1, 2, 4 } and place them along a circle as (1,4,2). Then the sums of adjacent numbers give all numbers from 1 to 7=3*(3-1)+1: { 1=1, 2=2, 3=1+2, 4=4, 5=1+4, 6=2+4, 7=1+2+4 }. Since such set is unique, a(3) = 1.
%Y A058241 Cf. A000010.
%K A058241 nonn,more
%O A058241 1,4
%A A058241 _Naohiro Nomoto_, Jan 16 2001
%E A058241 More terms from Rustem Aidagulov (rustem53(AT)mail.ru), Sep 06 2005 and Jan 01 2006
%E A058241 a(21)-a(23) from _Zhao Hui Du_, Mar 17 2019
%E A058241 Edited by _Max Alekseyev_, Jul 23 2019