A113900 -id:A113900 - cp's OEIS frontend

A271501 Erroneous version of A113900.

1, 6, 255, 92263, 280864514

Offset: 1

dead

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

A064941 Quartering a 2n X 2n chessboard (reference A257952) considering only the 90-deg rotationally symmetric results (omitting results with only 180-deg symmetry).

Original entry on oeis.org

1, 3, 26, 596, 38171, 7083827, 3852835452, 6200587517574, 29752897658253125, 427721252609771505989, 18479976131829456895423324, 2405174963192312814001570260392, 944597040906414962273553855513194341, 1120924326970482645724785944664901286951323

Offset: 1

Walter Gilbert (Walter(AT)Gilbert.net), Oct 28 2001

nonn

Walter Gilbert, Chessboard quartering; includes generating program.

Cf. A257952, A113900.

No formula known. However, the subset of solutions consisting of "tiles" with minimum edge lengths from a corner of the board to the center is A001700.

This sequence can be computed by counting paths in a graph. To compute the n-th term a graph with n X (n-1) vertices is required. Each graph vertex corresponds to 4 intersections between grid lines on the chessboard and graph edges correspond to ways of cutting the board along the grid lines. Frontier (matrix-transfer) graph path counting methods can then be applied to the graph to get the actual count. - Andrew Howroyd, Apr 18 2016

a(7)-a(8) from Juris Cernenoks, Feb 27 2013

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

A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

Original entry on oeis.org

0, 6, 53, 627, 16213, 1123743, 221984391, 127561384993, 215767063451331, 1082828220389781579, 16209089366362071416785, 726438398002211876667379681, 97741115155002465272674416929195, 39565596445488219947994403962984729307

Offset: 1

R. H. Hardin, Mar 02 2002

nonn

One of the partitions may completely surround the other. (Cf. A271802) - Andrew Howroyd, Apr 14 2016

Number of minimal edge cuts in the n X n grid graph. - Andrew Howroyd, Dec 11 2024

			Illustration of a(2)=6:
   11   12   12   12   11   11
   22   12   22   11   12   21
Illustration of a few solutions of a(3):
   111   112   122   111   111
   121   111   112   212   111
   111   111   222   222   222

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 1..26
Benjamin Fifield, Kosuke Imai, Jun Kawahara, and Christopher T. Kenny, The Essential Role of Empirical Validation in Legislative Redistricting Simulation, Tech. rep., Department of Government and Department of Statistics, Harvard University (2019).
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Minimal Edge Cut.

Cf. A068392, A166755, A271802, A068381, A068417, A113900, A348456, A358289.

a(n) = A271802(n) + A140517(n-2). - Andrew Howroyd, Apr 14 2016

a(n) = A166755(n)/2. - Andrew Howroyd, Dec 11 2024

a(7)-a(14) from Andrew Howroyd, Apr 14 2016

A271741 Number of ways to dissect a hexagon with side length n exactly into two identical parts in a triangular lattice.

Original entry on oeis.org

1, 8, 731, 982648, 16305532683, 3722056510716702, 11931439930135002524767

Offset: 1

Andrew Howroyd, Apr 15 2016

G. P. Jelliss, Dissected Hexagons, The Games and Puzzles Journal, Issue 22, January-April 2002.

Cf. A268606, A271857, A113900.

A271857 Number of ways to dissect a hexagon with side length n exactly into six identical parts in a triangular lattice.

Original entry on oeis.org

1, 2, 12, 173, 5429, 392544, 66961869, 27094069322, 26124568587557, 60352331499840380, 335377713005955826349, 4494480789037552980419332, 145516206571394421594063628243, 11398373584242623552596178870957640, 2162546126021822830176241418936795142991

Offset: 1

Andrew Howroyd, Apr 15 2016

nonn

This sequence only enumerates those dissections in which the six pieces are arranged around the center at 60-degree intervals. Other dissections are possible, such as those formed from those trisections which can be further bisected into identical parts.

G. P. Jelliss, Dissected Hexagons, The Games and Puzzles Journal, Issue 22, January-April 2002.

Cf. A113900, A268606, A271741.

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

A271858 Number of ways to trisect a triangle with side length n exactly into three identical parts in a triangular lattice.

Original entry on oeis.org

0, 1, 1, 2, 5, 20, 56, 276, 2136, 13756, 148352, 2727448, 41044816, 1056334024, 46033137324

Offset: 1

Andrew Howroyd, Apr 15 2016

In the case that n is not divisible by 3 the central triangle is removed, otherwise the dissection would not be possible.

G. P. Jelliss, Trisected Triangles, The Games and Puzzles Journal, Issue 22, January-April 2002.

Cf. A268606, A271857, A113900, A257952.

cp's OEIS Frontend

A271501 Erroneous version of A113900.

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

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A064941 Quartering a 2n X 2n chessboard (reference A257952) considering only the 90-deg rotationally symmetric results (omitting results with only 180-deg symmetry).

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A068416 Number of partitionings of n X n checkerboard into two edgewise-connected sets.

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A271741 Number of ways to dissect a hexagon with side length n exactly into two identical parts in a triangular lattice.

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A271857 Number of ways to dissect a hexagon with side length n exactly into six identical parts in a triangular lattice.

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

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A271858 Number of ways to trisect a triangle with side length n exactly into three identical parts in a triangular lattice.

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