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%I M0929 #53 Oct 07 2018 17:15:14
%S 1,1,0,0,2,4,0,2,10,32,38,140,496,1186,3178,16792,82038,289566,
%T 1139874,5914118,33800010,142337180,721286448,4384569864
%N Number of solutions to non-attacking reflecting queens problem.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Jordan Bell, Brett Stevens, <a href="https://doi.org/10.1016/j.disc.2007.12.043">A survey of known results and research areas for n-queens</a>, Discrete Mathematics, Volume 309 (2009), pp 1-31.
%H M. Gardner, <a href="http://www.logic-books.info/sites/default/files/k14-fractal_music_hypercard_and_more.pdf">Fractal Music, Hypercards and More</a>, Freeman, NY, 1991, p. 240.
%H G. B. Huff, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa23/aa2322.pdf">On pairings of the first 2n natural numbers</a>, Acta. Arith. 23 (1973) 117-126.
%H D. A. Klarner, <a href="https://www.jstor.org/stable/2315273">The Problem of Reflecting Queens</a>, The American Mathematical Monthly, Vol. 74, No. 8 (Oct., 1967), pp. 953-955.
%H M. Slater, <a href="http://dx.doi.org/10.1090/S0002-9904-1963-10907-6">Number theory Research Problem 1</a>, Bull. Amer. Math. Soc. 69 (1963), 333.
%e 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.
%o (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
%Y Cf. A000170, A002968, A051223, A051224, A272363.
%K nonn,nice,more
%O 0,5
%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
%E a(18)-a(21) from _Sean A. Irvine_, Jan 13 2018
%E a(0)-a(3) prepended by _Michel Marcus_, Oct 03 2018
%E a(22) from _Sean A. Irvine_, Oct 04 2018
%E a(23) from _Sean A. Irvine_, Oct 07 2018
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