A348456 -id:A348456 - cp's OEIS frontend

A007764 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid.

1, 2, 12, 184, 8512, 1262816, 575780564, 789360053252, 3266598486981642, 41044208702632496804, 1568758030464750013214100, 182413291514248049241470885236, 64528039343270018963357185158482118, 69450664761521361664274701548907358996488

Offset: 1

David Radcliffe and Don Knuth

The length of the path varies.

			Suppose we start at (1,1) and end at (n,n). Let U, D, L, R denote steps that are up, down, left, right.
a(2) = 2: UR or RU.
a(3) = 12: UURR, UURDRU, UURDDRUU, URUR, URRU, URDRUU and their reflections in the x=y line.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-338.
Guttmann A J and Jensen I 2022 Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices Journal of Physics A: Mathematical and Theoretical 55 012345, (33pp) ; arXiv:2208.06744, Aug 2022.
D. E. Knuth, 'Things A Computer Scientist Rarely Talks About,' CSLI Publications, Stanford, CA, 2001, pages 27-28.
D. E. Knuth, The Art of Computer Programming, Section 7.1.4.
Shin-ichi Minato, The power of enumeration - BDD/ZDD-based algorithms for tackling combinatorial explosion, Chapter 3 of Applications of Zero-Suppressed Decision Diagrams, ed. T. Satsoa and J. T. Butler, Morgan & Claypool Publishers, 2014
Shin-ichi Minato, Counting by ZDD, Encyclopedia of Algorithms, 2014, pp. 1-6.
Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).

I. Jensen, H. Iwashita, and R. Spaans, Table of n, a(n) for n = 1..27 (I. Jensen computed terms 1 to 20, H. Iwashita computed 21 and 22, R. Spaans computed 23 to 25, and H. Iwashita computed 26 and 27)
M. Bousquet-Mélou, A. J. Guttmann and I. Jensen, Self-avoiding walks crossing a square, arXiv:cond-mat/0506341 [cond-mat.stat-mech], 2005.
Doi, Maeda, Nagatomo, Niiyama, Sanson, Suzuki, et al., Time with class! Let's count! [Youtube-animation demonstrating this sequence. In Japanese with English translation]
Steven R. Finch, Self-Avoiding-Walk Connective Constants
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
H. Iwashita, J. Kawahara, and S. Minato, ZDD-Based Computation of the Number of Paths in a Graph, TCS Technical Report, TCS-TR-A-12-60, Hokkaido University, September 18, 2012.
H. Iwashita, Y. Nakazawa, J. Kawahara, T. Uno, and S. Minato, Efficient Computation of the Number of Paths in a Grid Graph with Minimal Perfect Hash Functions, TCS Technical Report, TCS-TR-A-13-64, Hokkaido University, April 26, 2013.
I. Jensen, Series Expansions for Self-Avoiding Walks
Shin-ichi Minato, Power of Enumeration - Recent Topics on BDD/ZDD-Based Techniques for Discrete Structure Manipulation, IEICE Transactions on Information and Systems, Volume E100.D (2017), Issue 8, p. 1556-1562.
Shin-ichi Minato, Motivating Problems and Algorithmic Solutions, Algorithmic Foundations for Social Advancement, Springer, Singapore (2025), 17-29.
OEIS Wiki, Self-avoiding_walks
Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda, and Isao Tanaka, Derivative structure enumeration using binary decision diagram, arXiv:2002.12603 [physics.comp-ph], 2020.
Ruben Grønning Spaans, C program
Jens Weise, Evolutionary Many-Objective Optimisation for Pathfinding Problems, Ph. D. Dissertation, Otto von Guercke Univ., Magdeburg (Germany, 2022), see p. 53.
Eric Weisstein's World of Mathematics, Self-Avoiding Walk

Main diagonal of A064298.

Cf. A064297, A271507, A001184, A000532.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A007764(n):
    if n == 1: return 1
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n * n
    paths = GraphSet.paths(start, goal)
    return paths.len()
print([A007764(n) for n in range(1, 10)])  # Seiichi Manyama, Mar 21 2020

Computed to n=12 by John Van Rosendale in 1981
Extended to n=13 by Don Knuth, Dec 07 1995
Extended to n=20 by Mireille Bousquet-Mélou, A. J. Guttmann and I. Jensen
Extended to n=22 using ZDD technique based on Knuth's The Art of Computer Programming (exercise 225 in 7.1.4) by H. Iwashita, J. Kawahara, and S. Minato, Sep 18 2012
Extended to n=25 using state space compression (with rank/unrank) and dynamic programming (based in I. Jensen) by Ruben Grønning Spaans, Feb 22 2013
Extended to n=26 by Hiroaki Iwashita, Apr 11 2013
Extended to n=27 by Hiroaki Iwashita, Nov 18 2013

A333323 Number of self-avoiding closed paths on an n X n grid which pass through NW and SE corners.

Original entry on oeis.org

1, 3, 42, 1799, 232094, 92617031, 115156685746, 442641690778179, 5224287477491915786, 188825256606226776728029, 20879416139356164466643759334, 7057757437924198729598570424130207, 7287699030020917172151307665469211016474, 22973720258279267139936821063450448822110219653

Offset: 2

Seiichi Manyama, Mar 23 2020

nonn

			a(2) = 1;
   +--*
   |  |
   *--+
a(3) = 3;
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *--*  *   *     *   *  *--*
      |  |   |     |   |     |
      *--+   *--*--+   *--*--+

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 2..27
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D2 (with offset 1).
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.

Cf. A007764, A333246, A333247, A333466.

Cf. A121785, A356610-A356616, A354511.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333323(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n * n)
    return cycles.len()
print([A333323(n) for n in range(2, 10)])

a(11) from Seiichi Manyama, Apr 07 2020

a(10) and a(12)-a(15) from Vaclav Kotesovec, Aug 16 2022 (computed by Anthony Guttmann)

A348452 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area.

Original entry on oeis.org

1, 1, 2, 0, 1, 1, 0, 10, 0, 0, 0, 0, 0, 1, 1, 70, 0, 117, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 80518, 264500, 442791, 0, 451206, 0, 0, 178939, 0, 0, 80092, 0, 0, 0, 0, 0, 6728, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

N. J. A. Sloane, Oct 27 2021

The board has n^2 squares. The colors do not matter. T(n,k) is zero unless k divides n^2. The tiles are rook-connected polygons made from n^2/k squares.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
A348453 (the main entry for this problem) displays the same data in a more compact way (by omitting the zero entries from each row).
The data is taken from A004003, A172477, and Schutzman & MGGG (2018).

			The first seven rows of the triangle are:
1,
1, 2, 0, 1,
1, 0, 10, 0, 0, 0, 0, 0, 1,
1, 70, 0, 117, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 80518, 264500, 442791, 0, 451206, 0, 0, 178939, 0, 0, 80092, 0, 0, 0, 0, 0, 6728, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
...
The domino is the only polyomino of area 2, and the 36 ways to tile a 4 X 4 square with dominoes are shown in one of the links.

Moon Duchin, Graphs, Geometry and Gerrymandering”, Talk at Celebration of Mind Conference, Oct 23 2021.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Zach Schutzman and MGGG, The Known Sizes of Grid Metagraphs, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.
N. J. A. Sloane, Illustration for T(3,3) = 10
N. J. A. Sloane, Illustration for T(4,2) = 70 [Labels give code, B = length of internal boundary, C = number of internal corners, G = group order, # = number of this type. Note that (B,C) determines the type]
N. J. A. Sloane, Illustration for T(4,8) = 36 [Slide from an old talk of mine]
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).

Cf. A348453. A348454 and A348455 are similar triangles with the data in each row reversed. The row sums are in A348789.

See also A004003, A099390, A172477, A348456.

A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.

More than the usual number of terms are given, in order to show the first seven rows.

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

A348453 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 10, 1, 1, 70, 117, 36, 1, 1, 4006, 1, 1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1, 1, 158753814, 1, 7157114189

Offset: 1

N. J. A. Sloane, Oct 27 2021

The board has n^2 squares. The colors do not matter. The tiles are rook-connected polygons made from n^2/d_k squares.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
A348452 displays the same data in a less compact way. The present triangle is obtained by omitting the zero entries from A348452.
The data is taken from A004003, A172477, A348456, and Schutzman & MGGG (2018).
T(8,2) = 7157114189 (see A348456). T(8,3) is presently unknown.

			The first eight rows of the triangle are:
  1,
  1, 2, 1,
  1, 10, 1,
  1, 70, 117, 36, 1,
  1, 4006, 1,
  1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1,
  1, 158753814, 1,
  1, 7157114189, ?, 187497290034, ?, ?, 1,
  ...
The corresponding divisors d_k are:
  1,
  1, 2, 4,
  1, 3, 9,
  1, 2, 4, 8, 16,
  1, 5, 25,
  ...
The domino is the only polyomino of area 2, and the 36 ways to tile a 4 X 4 square with dominoes are shown in one of the links.

Moon Duchin, Graphs, Geometry and Gerrymandering”, Talk at Celebration of Mind Conference, Oct 23 2021.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Zach Schutzman and MGGG, The Known Sizes of Grid Metagraphs, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.
N. J. A. Sloane, Illustration for T(3,2) = 10
N. J. A. Sloane, Illustration for T(4,2) = 70 [Labels give code, B = length of internal boundary, C = number of internal corners, G = group order, # = number of this type. Note that (B,C) determines the type]
N. J. A. Sloane, Illustration for T(4,4) = 36 [Slide from an old talk of mine]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.

Cf. A348452. A348454 and A348455 are similar triangles with the data in each row reversed.
See also A004003, A099390, A172477, A348456.
Cf. A048691 (row lengths).

A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.

T(8,2) added May 04 2022 (see A348456) - N. J. A. Sloane, May 05 2022

A348454 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with rook-connected polyominoes of area k.

Original entry on oeis.org

1, 1, 2, 0, 1, 1, 0, 10, 0, 0, 0, 0, 0, 1, 1, 36, 0, 117, 0, 0, 0, 70, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6728, 80092, 178939, 0, 451206, 0, 0, 442791, 0, 0, 264500, 0, 0, 0, 0, 0, 80518, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

N. J. A. Sloane, Oct 27 2021

This is an essentially identical triangle to A348452, except that the data in each row has effectively been reversed. Rather than copying everything here, please refer to A348452 for further information.

			Triangle begins:
1,
1, 2, 0, 1,
1, 0, 10, 0, 0, 0, 0, 0, 1,
1, 36, 0, 117, 0, 0, 0, 70, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 6728, 80092, 178939, 0, 451206, 0, 0, 442791, 0, 0, 264500, 0, 0, 0, 0, 0, 80518, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
...

Cf. A348452, A348453, A348455, and A348456.

More than the usual number of terms are given, in order to show the first seven rows.

A348455 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 10, 1, 1, 70, 117, 36, 1, 1, 4006, 1, 1, 6728, 80092, 178939, 451206, 442791, 264500, 80518, 1, 1, 158753814, 1

Offset: 1

N. J. A. Sloane, Oct 27 2021

This is an essentially identical triangle to A348453, except that the data in each row has effectively been reversed. Rather than copying everything here, please refer to A348453 for further information.

			Triangle begins:
1,
1, 2, 1,
1, 10, 1,
1, 70, 117, 36, 1,
1, 4006, 1,
1, 6728, 80092, 178939, 451206, 442791, 264500, 80518, 1,
1, 158753814, 1,
1, ?, ?, 187497290034, ?, 7157114189, 1,
...

Cf. A348452, A348453, A348454, A348456.

Cf. A048691 (row lengths).

A167242 Number of ways to partition a 2*n X 3 grid into 2 connected equal-area regions.

Original entry on oeis.org

1, 3, 19, 85, 355, 1435, 5717, 22645, 89521, 353735, 1397863, 5525341, 21846421, 86403027, 341822335, 1352660761, 5354124895, 21197945407, 83945924393, 332507403625, 1317329758675, 5220055148883, 20688989887169, 82013159349085, 325165555406795, 1289434099001055, 5114044079094817, 20286061330030705, 80481556028898031

Offset: 0

R. H. Hardin, Oct 31 2009

nonn

			Some solutions for n=4
...1.1.1...1.1.1...1.1.2...1.1.2...1.1.2...1.1.1...1.1.1...1.1.1...1.1.1
...1.1.1...1.1.2...1.2.2...1.1.2...1.2.2...2.2.1...1.1.1...2.1.1...1.1.1
...2.2.1...1.2.2...1.1.2...1.2.2...1.2.2...2.2.1...2.1.1...2.2.1...2.1.1
...2.1.1...1.2.2...1.2.2...1.2.2...1.1.2...2.2.1...2.2.1...2.1.1...2.2.1
...2.2.1...1.2.2...1.1.2...1.2.2...1.1.2...2.1.1...2.2.1...2.2.1...2.2.1
...2.2.1...1.1.2...1.1.2...1.2.2...1.1.2...2.1.1...2.1.1...2.1.1...2.2.1
...2.2.1...1.2.2...1.2.2...1.1.2...1.1.2...2.1.1...2.2.2...2.1.2...2.2.1
...2.2.2...1.2.2...1.2.2...1.1.2...2.2.2...2.2.2...2.2.2...2.2.2...2.2.2

D. E. Knuth (Proposer) and Editors (Solver), Balanced tilings of a rectangle with three rows, Problem 11929, Amer. Math. Monthly, 125 (2018), 566-568.

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.

Cf. A000045, A167243.

The solution to the Knuth problem gives an explicit g.f. and an explicit formula for a(n) in terms of Fibonacci numbers. - N. J. A. Sloane, May 25 2018

a(0) = 1 prepended by Don Knuth, May 11 2016

Terms a(21) and beyond from Roberto Tauraso, Oct 11 2016

A358289 Generalized Gerrymander sequence: number of ordered ways to divide an n X n square into two connected regions, both of area n^2/2 if n is even, or of areas (n^2-1)/2 and (n^2+1)/2 if n is odd.

Original entry on oeis.org

0, 4, 16, 140, 2804, 161036, 27803749, 14314228378, 21838347160809, 99704315229167288, 1367135978051264146578, 56578717186086829451888706, 7065692298178203128922479762418, 2670113158846160742372913777087464324, 3052313665715695874527667027409186333152556

Offset: 1

N. J. A. Sloane, Nov 25 2022

nonn

			For n = 2, the square can be split vertically or horizontally, and then there are two ways to order the regions, so a(2) = 2*2 = 4.
For n = 3 we must choose a connected region of area 4 with a connected complement of area 5.
The possibilities are
  XXO      XXX      XXX
  XXO      XOO      OXO
  OOO      OOO      OOO
  4 ways   8 ways   4 ways
so a(3) = 16.

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 1..17
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.

Cf. A348456.

a(2*n)/2 = A348456(n).

A359146 Divide a square into n similar rectangles; a(n) is the number of different proportions that are possible.

Original entry on oeis.org

1, 1, 3, 11, 51, 245, 1372

Offset: 1

N. J. A. Sloane, Feb 07 2023

Only the proportions of the rectangles are counted, not how the rectangles are arranged in the square.

The number b(n) of different ways to divide a square into n similar rectangles, up to rotation and reflection, is at least a(n). We have b(1)=1, b(2)=1, b(3)=3, and Baez remarks that b(4) > 11. It would be nice to know more. Is the sequence {b(n)} already in the OEIS?

Siobhan Roberts, An Online Puzzle Excites Math Fans, New York Times, Feb 07 2023, pages D1 and D4.

John Baez, Rahul Narain, Daniel Piker, Lisanne Taams, Ian Henderson, David Eppstein, and others, Dividing a Square into Similar Rectangles, First posted Dec. 22 2022, with many later comments
John Baez, Dividing a Square into 7 Similar Rectangles, Mar 06 2023. Continuation of first blog entry, with a new value for a(7).
Ian Henderson, Cutting squares into similar rectangles using a computer program, Feb 02 2023
Siobhan Roberts, The Quest to Find Rectangles in a Square, New York Times, Feb 07 2023
N. J. A. Sloane, Illustration for a(3) = 3

Cf. A100664, A108066, A348456.

It appears that a(8) = 8522. - Ian Henderson, Feb 07 2023, corrected Mar 07 2023

a(7) corrected by N. J. A. Sloane, Mar 07 2023, based on the second John Baez blog entry.

cp's OEIS Frontend

A007764 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid.

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A333323 Number of self-avoiding closed paths on an n X n grid which pass through NW and SE corners.

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A348452 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area.

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

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A348453 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.

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A348454 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with rook-connected polyominoes of area k.

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A348455 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with rook-connected polyominoes of area d_k, where d_k is the k-th divisor of n^2.

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A167242 Number of ways to partition a 2*n X 3 grid into 2 connected equal-area regions.

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A358289 Generalized Gerrymander sequence: number of ordered ways to divide an n X n square into two connected regions, both of area n^2/2 if n is even, or of areas (n^2-1)/2 and (n^2+1)/2 if n is odd.

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