1,2

The three sequences A054500/A054501/A054502 are used to classify solutions to the problem of "Nonattacking queens on a 2n+1 X 2n+1 toroidal board" by their symmetry; solutions are considered equivalent iff they differ only by rotation, reflection or torus shift.

For brevity, let i(n) = A054500(n) (indicator sequence), m(n) = A054501(n) (multiplicity) and c(n) = A054502(n) (count).

i(n) = k means that there are solutions for the kXk board and that m(n) and c(n) refer to it. There are c(n) inequivalent solutions which may be extended to m(n) different representations each (i.e. m(n) permutations).

This gives two formulae: A007705(n) = Sum (c(k) * m(k)), A053994(n) = Sum (c(k)), where the sum is taken over all k for which i(k) = 2n+1, for both formulae. Note that m(n) is always a divisor of 8 * i(n)^2.

I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math.Monthly, 101 (1994), 629-639 (for finding the solutions).

A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc.Ninth Australian Conference (Brisbane, August 1981). Ed. E. J.Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math.,952. Springer-Verlag, 1982 (for getting equivalence classes).

For a 19 X 19 toroidal board, you have three entries in the indicator sequence A054500; their count terms (A054502) give 354 = 4 + 132 + 218 inequivalent solutions; together with their multiplicity (A054501) they add up to 4 * 76 + 132 * 1444 + 218 * 2888 = 820496 solutions at all.

Cf. A054501, A054502, A053994, A007705, A006841.

Sequence in context: A095798 A136142 A136162 this_sequence A082684 A096379 A098761

Adjacent sequences: A054497 A054498 A054499 this_sequence A054501 A054502 A054503

nonn,nice,hard

Matthias Engelhardt (Matthias.R.Engelhardt(AT)web.de, Apr 10, 2000)

More terms from Matthias Engelhardt (Matthias.R.Engelhardt(AT)web.de), Jan 11 2001