A003213 -id:A003213 - cp's OEIS frontend

A257952 Number of ways to quarter a 2n X 2n chessboard.

1, 1, 5, 37, 766, 43318, 7695805, 4015896016, 6371333036059, 30153126159555641, 431453249608567040694, 18558756256964594960321428, 2411839397220672351872242339314, 945878376319424018440202856702995909, 1121914029089423867715407724741780046405923

Offset: 0

Giovanni Resta, May 14 2015

nonn

Number of ways to dissect a 2n X 2n chessboard into 4 congruent pieces. As stated by Thomas R. Parkin in his letter (see Links), the dissections belong to two classes. One in which the cuts divide the chessboard into four pieces which are 90-degree rotationally symmetric, the other in which the square is first bisected in two rectangles and then each rectangle is divided into two pieces which are 180-degree rotationally symmetric.

Two dissections are considered distinct if they belong to two different classes, even if the tile is the same. In both classes reflections and rotations are not counted, and moreover in the second class two dissections are considered the same if they differ only by the orientation of the tiles.

M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
Popular Computing (Calabasas, CA), Vol. 1 (No. 7, 1973), Problem 15, front cover and page 2.

T. R. Parkin, Letter to N. J. A. Sloane, Feb 01, 1974. This letter contained as an attachment the following 11-page letter to Fred Gruenberger.
T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974
T. R. Parkin, Discussion of Problem 15, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), pages PC15-4 to PC15-8.
Popular Computing (Calabasas, CA), Illustration showing that a(3) = 37, Vol. 1 (No. 7, 1973), front cover. (One of the 37 is simply the square divided into four quadrants.)
Giovanni Resta, Illustration of a(4) = 766.

Cf. A003213 (another version, but probably incorrect - N. J. A. Sloane, Apr 17 2016), A006067, A064941, A113900, A268606.

A006067 = Import["https://oeis.org/A006067/b006067.txt", "Table"][[All, 2]];
a[n_] := If[n == 0, 1, A006067[[2n]]];
a /@ Range[0, 14] (* Jean-François Alcover, Sep 14 2019 *)

a(n) = A006067(2n) for n>0. - Jean-François Alcover, Sep 14 2019, after Andrew Howroyd in A006067.

a(9)-a(14) from Andrew Howroyd, Apr 18 2016

A006067 Number of ways to quarter an n X n chessboard, with the central square removed for odd n.

Original entry on oeis.org

1, 1, 1, 5, 7, 37, 104, 766, 3970, 43318, 431932, 7695805, 137066448, 4015896016, 128095791922, 6371333036059, 355704307903818, 30153126159555641, 2952926822418475378, 431453249608567040694, 73569487283165427567144, 18558756256964594960321428

Offset: 1

N. J. A. Sloane

To "quarter" means to dissect in 4 parts, identical up to rotation, whose interior must be connected. (I.e., the parts must be polyominoes, any 1 X 1 square of which must share a side with some other 1 X 1 square of the part, unless there's only one.) Solutions that differ only by rotation or reflection are not counted separately.
See A257952 for much more information.
See A272070 for information on odd terms.

			For n = 1, we have the 1 X 1 board of which we remove the central square, so nothing is left, and the empty tiling is the only possible tiling.
For n = 2, we have the 2 X 2 board which can only be quartered using four 1 X 1 squares, so a(2) = 1 as well.
For n = 3, the 3 X 3 board without the central square can only be quartered using four 2 X 1 rectangles, so a(3) = 1 as well. (The two different solutions where the top rectangle is aligned to the left or to the right are counted as one, since they only differ by reflection.)
For n = 4 there is the trivial solution using squares, one using straight 4 X 1 tiles, one using T-shaped tiles, and two non-isomorphic ones using L-shaped tiles, one with a central symmetry and one with an axial symmetry:
          A A B B       A B C D       A B B B        A A B B        A A B B
  square: A A B B   I:  A B C D   T:  A A B C   Lc:  A C B D   La:  A C D B
          C C D D       A B C D       A D C C        A C B D        A C D B
          C C D D       A B C D       D D D C        C C D D        C C D D

M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
T. R. Parkin, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..28
Audrey Gruenberger (Editor), Checkerboard - Problem 15, Popular Computing, Vol. 1, No. 7 (1973), front cover and page 2. (Local copy of a scan of the cover illustration showing the a(6)-1 = 36 nontrivial solutions for n = 6, omitting the trivial solution using four squares.)

Cf. A257952, A064941, A272070.

a(2n) = A257952(n), a(2n+1) = A272070(n). - Andrew Howroyd, Apr 19 2016

a(8) corrected, a(9)-a(22) from Andrew Howroyd, Apr 18 2016

Name edited to clarify definition for odd n, and other edits by M. F. Hasler, Jun 13 2025

A003155 Number of ways to halve an n X n chessboard.

Original entry on oeis.org

1, 1, 1, 6, 15, 255, 1897, 92263, 1972653, 281035054, 17635484470, 7490694495750, 1405083604458437, 1789509008288411290

Offset: 1

N. J. A. Sloane

This is the number of ways to cut an n X n chessboard along grid lines into two identical pieces. In the case that n is odd the central square is first removed to ensure an even number of squares. Solutions that differ only by rotation or reflection are considered equivalent. - Andrew Howroyd, Apr 19 2016

M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Illustration showing ways to quarter a 6x6 board

Cf. A113900, A006067, A257952, A271741.

a(2n) = A113900(n). - Andrew Howroyd, Apr 19 2016

a(10) corrected and a(11)-a(14) from Andrew Howroyd, Apr 19 2016

cp's OEIS Frontend

A257952 Number of ways to quarter a 2n X 2n chessboard.

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A006067 Number of ways to quarter an n X n chessboard, with the central square removed for odd n.

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A003155 Number of ways to halve an n X n chessboard.

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