A007631 Number of solutions to non-attacking reflecting queens problem. (Formerly M0929)

1, 1, 0, 0, 2, 4, 0, 2, 10, 32, 38, 140, 496, 1186, 3178, 16792, 82038, 289566, 1139874, 5914118, 33800010, 142337180, 721286448, 4384569864, 28417161758, 163597795748

Offset

0,5

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+xi and yi-xi are all different. - Michel Marcus, Apr 27 2016 [Corrected by Martin Fuller, Nov 29 2025]

References

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

Links

Table of n, a(n) for n=0..25.

Jordan Bell and Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309 (2009), pp 1-31.

Martin 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.

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A256280 A153182 A111818 * A281765 A155517 A325490

Adjacent sequences: A007628 A007629 A007630 * A007632 A007633 A007634

Keyword

nonn,nice,more

Author

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

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

a(24) from Martin Fuller, Nov 29 2025

a(25) from Martin Fuller, Nov 30 2025

Status

approved