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A348456 Number of ways to dissect a 2*n X 2*n chessboard into two polyominoes each of area 2*n^2. 12
1, 2, 70, 80518, 7157114189, 49852157614583644, 28289358593043414725944353, 1335056579423080371186456888543732162, 5288157175943649955880910966508435029578848399795, 1768514227824943648668138153226998430209626836775021539911012000, 50126261987194138333095266040242179892262270498222242227767710277119489194126252, 120727080026653995683405108506109122788592972611035310673809853406496349171003311517916839962975062 (list; graph; refs; listen; history; text; internal format)
See A348453 for much more information.
The board has 4*n^2 squares. The colors of the squares do not matter. The two parts are rook-connected polygons of area 2*n^2. They do not need to be the same polygon, only that they have the same area.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
a(4) was found on May 04 2022 by George Spahn and Manuel Kauers using an 1838 X 1838 transfer matrix found by George Spahn (see the Zeilberger link). Manuel Kauers computed the [1,2] entry of the 9th power of that matrix. The desired number a(4) is half of the coefficient of z^32 in that entry. - Doron Zeilberger, May 04 2022
Also known as the "Gerrymander Sequence" per Kauers, et al. - Michael De Vlieger, Dec 06 2022
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
Manuel Kauers, Christoph Koutschan, and George Spahn, A348456(4) = 7157114189, arXiv:2209.01787 [math.CO], 2022.
Manuel Kauers, Christoph Koutschan, and George Spahn, How Does the Gerrymander Sequence Continue?, J. Int. Seq., Vol. 25 (2022), Article 22.9.7.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences: An illustrated guide with many unsolved problems, Guest Lecture given in Doron Zeilberger's Experimental Mathematics Math640 Class, Rutgers University, Spring Semester, Apr 28 2022: Slides; Slides (an alternative source).
A column of A348452 and A348453, and a diagonal of A348454 and A348455.
See also A358289.
Cf. A167242.
Sequence in context: A132566 A151686 A201555 * A358275 A293753 A164554
N. J. A. Sloane, Oct 27 2021
Added a(5)-a(7) (from the Kauers et al. reference), Joerg Arndt, Sep 07 2022
a(8)-a(11) from Guttmann and Jensen (2022).
a(0)=1 prepended by Alois P. Heinz, Dec 06 2022

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Last modified December 3 08:55 EST 2023. Contains 367531 sequences. (Running on oeis4.)