OFFSET
0,3
COMMENTS
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.
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]
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
LINKS
Rintaro Matsuo, Table of n, a(n) for n = 0..600 (terms up to a(35) from Vaclav Kotesovec)
Manuel Kauers, Guessed recurrence operator of order 24 and degree 64
Vaclav Kotesovec, Mathematica program for this sequence
George Spahn and Doron Zeilberger, Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s), arXiv:2211.02550 [math.CO], 2022.
Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS, vol. 6 (2006), paper A11. arXiv:math/0507293.
FORMULA
A formula is given in the Tauraso reference.
Asymptotic (R. Tauraso 2006, quadratic term V. Kotesovec 2011): a(n)/n! ~ (1 + 4/n + 8/n^2)/e^2.
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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Roberto Tauraso, A. Nicolosi and G. Minenkov, Jul 13 2005
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Vladeta Jovovic, Jan 01 2008
Terms a(33)-a(35) from Vaclav Kotesovec, Apr 20 2012
STATUS
approved