A007631 Number of solutions to non-attacking reflecting queens problem.

1, 1, 0, 0, 2, 4, 0, 2, 10, 32, 38, 140, 496, 1186, 3178, 16792, 82038, 289566, 1139874, 5914118, 33800010, 142337180, 721286448, 4384569864

Offset: 0

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Keywords

• nonn
• nice
• more

Comments

a(n) is the number of ways to pair the natural numbers from 1 to n with those between n+1 and 2*n into n pairs (xi,yi) such that the 2*n numbers yi+i and yi-i are all different. - Michel Marcus, Apr 27 2016

Examples

			For n = 4, ((1,7), (2,5), (3,8), (4,6)) is an instance of such grouping. ((2,5), (1,7), (3,8), (4,6)) is considered to be the same grouping.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Jordan Bell, Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309 (2009), pp 1-31.
• M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, p. 240.
• G. B. Huff, On pairings of the first 2n natural numbers, Acta. Arith. 23 (1973) 117-126.
• D. A. Klarner, The Problem of Reflecting Queens, The American Mathematical Monthly, Vol. 74, No. 8 (Oct., 1967), pp. 953-955.
• M. Slater, Number theory Research Problem 1, Bull. Amer. Math. Soc. 69 (1963), 333.

Crossrefs

Cf. A000170, A002968, A051223, A051224, A272363.

Programs

PARI

a(n) = {nb = 0; for (j=0, n!-1, vp = numtoperm(n, j); vb = vector(n, k, vp[k]+n); vs = vector(n, k, vb[k]+k); vd = vector(n, k, vb[k]-k); if (#vs + #vd == #Set(concat(vs, vd)), nb++); ); nb; } \\ Michel Marcus,  Apr 27 2016

Extensions

a(18)-a(21) from Sean A. Irvine, Jan 13 2018

a(0)-a(3) prepended by Michel Marcus, Oct 03 2018

a(22) from Sean A. Irvine, Oct 04 2018

a(23) from Sean A. Irvine, Oct 07 2018