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

Author

Giovanni Resta, May 14 2015

Keywords

• nonn

Comments

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.

References

• 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.

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

Extensions

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