1,6

The position of weight 1 is kept fixed at position 1. Mirror configurations are counted only once. Proposed in the seqfan mailing list by Brendan D. McKay (bdm(AT)cs.anu.edu.au), Sep 12 2005. Also number of permutations p1,p2,...,pn such that the polynomial p1 + p2*x + ... + pn*x^(n-1) has exp(2*pi*i/n) as a zero. Also number of equiangular polygons whose sides are some permutation of 1,2,3,...,n. T. D. Noe (noe(AT)sspectra.com), Sep 13 2005. No solutions exist if n is a prime power. Proved by Edwin Clark (eclark(AT)math.usf.edu), Sep 14 2005.

Hugo Pfoertner, Balanced weights on circle (Tables of configurations)

Bernoff's Puzzler, MuddMath Newsletter Volume 4, No. 1, Page 10, Spring 2005

The smallest n for which a solution exists is n=6 with 4 solutions:

...........Weight

......1..2..3..4..5..6

.Count...at.position

..1...1..4..5..2..3..6

..2...1..5..3..4..2..6

..3...1..6..2..4..3..5

..4...1..6..3..2..5..4

Configurations 1 is the mirror image of configuration 4, ditto for configurations 2 and 3. Therefore a(6)=2.

Needs["DiscreteMath`Combinatorica`"]; Table[eLst=E^(2.*Pi*I*Range[n]/n); Count[(Permutations[Range[n]]), q_List/; Chop[q.eLst]===0]/(2n), {n, 10}] (* very slow for n>10 *) - T. D. Noe (noe(AT)sspectra.com), May 05 2006

Cf. A118888 [Configurations with minimum imbalance], A063697 [Positions of positive coefficients in cyclotomic polynomial in binary], A063699 [Positions of negative coefficients in cyclotomic polynomial in binary].

Sequence in context: A063698 A136615 A029696 this_sequence A057383 A086260 A124505

Adjacent sequences: A118884 A118885 A118886 this_sequence A118888 A118889 A118890

hard,more,nonn

Hugo Pfoertner (hugo(AT)pfoertner.org), May 03 2006