Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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