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%I A257952 #25 Dec 15 2021 01:19:31
%S A257952 1,1,5,37,766,43318,7695805,4015896016,6371333036059,
%T A257952 30153126159555641,431453249608567040694,18558756256964594960321428,
%U A257952 2411839397220672351872242339314,945878376319424018440202856702995909,1121914029089423867715407724741780046405923
%N A257952 Number of ways to quarter a 2n X 2n chessboard.
%C A257952 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.
%C A257952 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.
%D A257952 M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
%D A257952 Popular Computing (Calabasas, CA), Vol. 1 (No. 7, 1973), Problem 15, front cover and page 2.
%H A257952 T. R. Parkin, <a href="/A257952/a257952.png">Letter to N. J. A. Sloane, Feb 01, 1974</a>. This letter contained as an attachment the following 11-page letter to Fred Gruenberger.
%H A257952 T. R. Parkin, <a href="/A257952/a257952.pdf">Letter to Fred Gruenberger, Jan 29, 1974</a>
%H A257952 T. R. Parkin, <a href="/A257952/a257952_1.pdf">Discussion of Problem 15</a>, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), pages PC15-4 to PC15-8.
%H A257952 Popular Computing (Calabasas, CA), <a href="/A257952/a257952.jpg">Illustration showing that a(3) = 37</a>, Vol. 1 (No. 7, 1973), front cover. (One of the 37 is simply the square divided into four quadrants.)
%H A257952 Giovanni Resta, <a href="/A257952/a257952_2.pdf">Illustration of a(4) = 766</a>.
%F A257952 a(n) = A006067(2n) for n>0. - _Jean-François Alcover_, Sep 14 2019, after _Andrew Howroyd_ in A006067.
%t A257952 A006067 = Import["https://oeis.org/A006067/b006067.txt", "Table"][[All, 2]];
%t A257952 a[n_] := If[n == 0, 1, A006067[[2n]]];
%t A257952 a /@ Range[0, 14] (* _Jean-François Alcover_, Sep 14 2019 *)
%Y A257952 Cf. A003213 (another version, but probably incorrect - _N. J. A. Sloane_, Apr 17 2016), A006067, A064941, A113900, A268606.
%K A257952 nonn
%O A257952 0,3
%A A257952 _Giovanni Resta_, May 14 2015
%E A257952 a(9)-a(14) from _Andrew Howroyd_, Apr 18 2016

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