A035290 -id:A035290 - cp's OEIS frontend

A294240 The number of possible ways in which 2n^2 black pawns and 2n^2 white pawns can be arranged on a 2n X 2n chessboard such that no pawn attacks another.

1, 3, 30, 410, 6148, 96120, 1526700, 24425026, 392143828, 6306613690, 101505099104, 1634209596410, 26311180850268, 423567557239604, 6817440328754244, 109703307312544664, 1764863031686159684, 28385338557467333804, 456426743658724223028, 7337464027218416593362

Offset: 0

Peter Kagey, Oct 25 2017

nonn

White pawns attack diagonally up and black pawns attack diagonally down.

En passant capturing is not possible.

			For n = 1 the a(1) = 3 boards are as follows:
  +---+---+    +---+---+    +---+---+
  | W | W |    | B | W |    | W | B |
  +---+---+    +---+---+    +---+---+
  | B | B |    | W | B |    | B | W |
  +---+---+    +---+---+    +---+---+
.
An example of one of the a(2) = 30 boards is:
  +---+---+---+---+
  | W | W | W | W |
  +---+---+---+---+
  | B | W | W | W |
  +---+---+---+---+
  | B | B | W | B |
  +---+---+---+---+
  | B | B | B | B |
  +---+---+---+---+

Code Golf Stack Exchange user feersum, Too many pawns on a chess board.

Cf. A035290.

A384759 Number of legal arrangements in pawn-only chess on an n X n board where no pieces have been taken and no piece attacks another piece.

Original entry on oeis.org

0, 3, 2031, 728174, 247646098, 91880342535, 38818192375310, 18907485764545412, 10626953883068264472, 6866760686250915376779, 5073038373153476636807709, 4259014676256866422905669602, 4038463837000965678262091166880, 4299625631242136963071149921577615, 5111407212497576694797045579672852791

Offset: 4

Edwin Hermann, Jun 09 2025

nonn

The number of ways of arranging n pawns of each color on an n X n board such that no pawn threatens another, each file contains one pawn of each color, none of the pawns are passed pawns, and each pawn is placed between row 2 and row n-1 inclusive.

There is no requirement that the arrangements counted here can actually be achieved via a sequence of legal chess moves.

			The a(5) = 3 positions are:
  . . . . .    . . . . .    . . . . .
  b b b b b    . b . b .    b . b . b
  . . . . .    b w b w b    w b w b w
  w w w w w    w . w . w    . w . w .
  . . . . .    . . . . .    . . . . .

Cf. A035290, A294240.

MkTfrMtx(n)={my(m=binomial(n,2), M=matrix(m,m)); for(i=1,n-1, for(j=i+1,n, for(p=1,n-1, for(q=p+1,n, if(q<>i+1&&j<>p+1, M[binomial(n-i,2)+(j-i), binomial(n-p,2)+(q-p)]=1) )))); M}
a(n)={my(M=MkTfrMtx(n-2)); vecsum(M^(n-1)*vectorv(#M,i,1))} \\ Andrew Howroyd, Jun 15 2025

a(9) onwards from Andrew Howroyd, Jun 15 2025

cp's OEIS Frontend

A294240 The number of possible ways in which 2n^2 black pawns and 2n^2 white pawns can be arranged on a 2n X 2n chessboard such that no pawn attacks another.

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A384759 Number of legal arrangements in pawn-only chess on an n X n board where no pieces have been taken and no piece attacks another piece.

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cp's OEIS Frontend

A294240 The number of possible ways in which 2*n^2 black pawns and 2*n^2 white pawns can be arranged on a 2n X 2n chessboard such that no pawn attacks another.

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A384759 Number of legal arrangements in pawn-only chess on an n X n board where no pieces have been taken and no piece attacks another piece.

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Author

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PARI

Extensions

A294240 The number of possible ways in which 2n^2 black pawns and 2n^2 white pawns can be arranged on a 2n X 2n chessboard such that no pawn attacks another.