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A118887
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Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk so that the disk will exactly balance on the center point.
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1
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0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 0, 732, 0, 720, 48, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| 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.
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LINKS
| Hugo Pfoertner, Balanced weights on circle (Tables of configurations)
Bernoff's Puzzler, MuddMath Newsletter Volume 4, No. 1, Page 10, Spring 2005
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EXAMPLE
| 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.
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MATHEMATICA
| 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
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CROSSREFS
| 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: A179677 A136615 A029696 * A057383 A169772 A193542
Adjacent sequences: A118884 A118885 A118886 * A118888 A118889 A118890
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KEYWORD
| hard,more,nonn
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AUTHOR
| Hugo Pfoertner (hugo(AT)pfoertner.org), May 03 2006
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