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%I #43 Feb 08 2023 09:07:07
%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<n-1.
%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 non-attacking 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]
%C Note that the conjectured recurrence was based on the 600-term b-file, not the other way round. - _N. J. A. Sloane_, Dec 07 2022
%H Rintaro Matsuo, <a href="/A110128/b110128.txt">Table of n, a(n) for n = 0..600</a> (terms up to a(35) from Vaclav Kotesovec)
%H Manuel Kauers, <a href="/A110128/a110128_1.txt">Guessed recurrence operator of order 24 and degree 64</a>
%H Vaclav Kotesovec, <a href="/A110128/a110128.txt">Mathematica program for this sequence</a>
%H George Spahn and Doron Zeilberger, <a href="https://arxiv.org/abs/2211.02550">Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s)</a>, arXiv:2211.02550 [math.CO], 2022.
%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.
%F a(n) ~ exp(-2) * n! * (1 + 4/n + 8/n^2 + 68/(3*n^3) + 242/(3*n^4) + 1692/(5*n^5) + 72802/(45*n^6) + 2725708/(315*n^7) + 16083826/(315*n^8) + 186091480/(567*n^9) + 32213578294/(14175*n^10) + ...), based on the recurrence by _Manuel Kauers_. - _Vaclav Kotesovec_, Dec 05 2022
%Y Cf. A089222, A002464, A117574, A189281.
%Y Column k=2 of A333706.
%K nonn,nice
%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
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