

A257952


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


9



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

0,3


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 degrees 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 degrees 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

Table of n, a(n) for n=0..14.
T. R. Parkin, Letter to N. J. A. Sloane, Feb 01, 1974. This letter contained as an attachment the following 11page 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 PC154 to PC158.
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.


FORMULA

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


MATHEMATICA

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


CROSSREFS

Cf. A003213 (another version, but probably incorrect  N. J. A. Sloane, Apr 17 2016), A006067, A064941, A113900, A268606.
Sequence in context: A215233 A083846 A180275 * A240186 A003213 A166851
Adjacent sequences: A257949 A257950 A257951 * A257953 A257954 A257955


KEYWORD

nonn


AUTHOR

Giovanni Resta, May 14 2015


EXTENSIONS

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


STATUS

approved



