%I
%S 1,1,2,4,16,44,200,1288,9512,78652,744360,7867148,91310696,1154292796,
%T 15784573160,232050062524,3648471927912,61080818510972,
%U 1084657970877416,20361216987032284,402839381030339816,8377409956454452732
%N Number of permutations p of 1,2,...,n satisfying p(i+2)p(i) not equal to 2 for all 0<i<n1.
%C When n is even: 1) Number of ways that n persons seated at a rectangular table with n/2 seats along the two opposite sides can be rearranged in such a way that neighbors are no more neighbors after the rearrangement. 2) Number of ways to arrange n kings on an n X n board, with 1 in each row and column, which are nonattacking with respect to the main four quadrants.
%C a(n) is also number of ways to place n nonattacking pieces rook + alfil on an n X n chessboard (Alfil is a leaper [2,2]) [From _Vaclav Kotesovec_, Jun 16 2010]
%H Vaclav Kotesovec, <a href="/A110128/b110128.txt">Table of n, a(n) for n = 0..35</a>
%H Vaclav Kotesovec, <a href="/A110128/a110128.txt">Mathematica program for this sequence</a>
%H Roberto Tauraso, <a href="http://www.emis.de/journals/INTEGERS/papers/g11/g11.pdf">The Dinner Table Problem: The Rectangular Case</a>, INTEGERS, vol. 6 (2006), paper A11. arXiv:<a href="http://arxiv.org/abs/math/0507293">math/0507293</a>.
%F A formula is given in the Tauraso reference.
%F Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 4/n + 8/n^2)/e^2.
%Y Cf. A089222, A002464, A117574.
%K nonn,nice,hard
%O 0,3
%A _Roberto Tauraso_, A. Nicolosi and G. Minenkov, Jul 13 2005
%E Edited by _N. J. A. Sloane_ at the suggestion of Vladeta Jovovic, Jan 01 2008
%E Terms a(33)a(35) from _Vaclav Kotesovec_, Apr 20 2012
