A103331 -id:A103331 - cp's OEIS frontend

A103330 Number of ways to place n+1 queens and a pawn on an n X n board so that no two queens attack each other.

0, 0, 0, 0, 0, 16, 20, 128, 396, 2288, 11152, 65712, 437848, 3118664, 23387448, 183463680, 1474699536, 12485203304, 110956890352, 1028589512656, 9801351322432, 97731300891440, 1014610719838792

Offset: 1

R. Douglas Chatham (d.chatham(AT)moreheadstate.edu), Jan 31 2005

			a(4) = 0 because when 5 queens are placed on a 4 X 4 board, at least 2 queens will be adjacent and therefore mutually attacking.

Hans Bodlaender, The Nine Queens Problem, posted 4 January 2004.
R. D. Chatham, The N+k Queens Problem Page.
R. D. Chatham, The N+k Queens Problem Page.
R. D. Chatham, M. Doyle, G. H. Fricke, J. Reitmann, R. D. Skaggs and M. Wolff, Independence and Domination Separation in Chessboard Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 68 (2008).
R. D. Chatham, G. H. Fricke and R. D. Skaggs, The Queens Separation Problem, Utilitas Mathematica 69 (2006), 129-141.

Cf. A000170, A103331.

Further terms from R. Douglas Chatham (d.chatham(AT)moreheadstate.edu), Feb 15 2005, Apr 20 2007, Apr 28 2007
a(12) corrected by R. Douglas Chatham (d.chatham(AT)moreheadstate.edu), May 12 2009
a(18)-a(21) from Martin Ehrenstein, Oct 24 2023
a(22) from Martin Ehrenstein, Feb 09 2024
a(23) computed on a GPU using CUDA by Martin Ehrenstein, Aug 10 2025

A368061 The number of ways to place one pawn and n+1 nonattacking queens on an n X n chessboard, unchanged by 180-degree rotation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 18, 0, 50, 0, 652, 0, 4260, 0, 25134, 0, 238098, 0, 2492304, 0, 25389294

Offset: 1

Don Knuth, Dec 10 2023

n must be odd, and the pawn must be in the center.

			For n=7 the a(7)=1 solution is
  . . Q . . . .
  . . . . . Q .
  . . . Q . . .
  Q . . P . . Q
  . . . Q . . .
  . Q . . . . .
  . . . . Q . .

A103330 counts all solutions; A103331 counts them with symmetry removed.

A103330(n) = 8*A103331(n) - 4*a(n).

cp's OEIS Frontend

A103330 Number of ways to place n+1 queens and a pawn on an n X n board so that no two queens attack each other.

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