A001184 -id:A001184 - 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

A120443 Number of (undirected) Hamiltonian paths in the n X n grid graph.

Original entry on oeis.org

1, 4, 20, 276, 4324, 229348, 13535280, 3023313284, 745416341496, 730044829512632, 786671485270308848, 3452664855804347354220, 16652005717670534681315580, 331809088406733654427925292528, 7263611367960266490262600117251524

Offset: 1

David Bevan, Jul 19 2006

			From _Robert FERREOL_, Apr 03 2019: (Start)
a(3) = 20:
there are 4 paths similar to
  + - + - +
          |
  + - + - +
  |
  + - + - +
8 paths similar to
  + - + - +
  |       |
  +   + - +
  |   |
  +   + - +
and 8 paths similar to
  + - + - +
  |       |
  +   +   +
  |   |   |
  +   + - +
(End)

Jesper L. Jacobsen, Table of n, a(n) for n = 1..17
J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A: Math. Theor. 40 (2007) 14667-14678.
J.-M. Mayer, C. Guez and J. Dayantis, Exact computer enumeration of the number of Hamiltonian paths in small square plane lattices, Physical Review B, Vol. 42 Number 1, 1990.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path
Index entries for sequences related to graphs, Hamiltonian

Main diagonal of A332307.
Cf. A003685, A003695, A003778, A145402.
Cf. A003763, A000532, A001184, A145157, A271507, A007764, A121785, A121789.

a(n) = A096969(n) / 2 for n > 1.

More terms from Jesper L. Jacobsen (jesper.jacobsen(AT)u-psud.fr), Dec 12 2007

A000532 Number of Hamiltonian paths from NW to SW corners in an n X n grid.

Original entry on oeis.org

1, 1, 2, 8, 86, 1770, 88418, 8934966, 2087813834, 1013346943033, 1111598871478668, 2568944901392936854, 13251059359839620127088, 145194816279817259193401518, 3524171261632305641165676374930, 182653259988707123426135593460533473

Offset: 1

Russ Cox, Mar 15 1996

Number of walks reaching each cell exactly once.

Oliver R. Bellwood, Table of n, a(n) for n = 1..20 (first 19 terms from KeyTo9(AT)Fans, Ed Wynn)
KeyTo9(AT)Fans, Counting paths in a grid - Chinese web page giving the sequence up to 18 items.
Douglas M. McKenna, Tendril Motifs for Space-Filling, Half-Domino Curves, in: Bridges Finland Conference Proceedings, 2016, pp. 119-126.
Douglas M. McKenna, Are Maximally Unbalanced Hilbert-Style Square-Filling Curve Motifs a Drawing Medium?, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 91-98.

Main diagonal of A271592.
Cf. A181688, A181689, A014524, A014585.
Cf. A001184, A145157, A120443, A003763, A271507, A007764, A121785, A121789.
See also A350148.

More terms from Zhao Hui Du, Jul 08 2008
Edited by Franklin T. Adams-Watters, Jul 03 2009
Name clarified by Andrew Howroyd, Apr 10 2016

A145157 Number of Greek-key tours on an n X n board; i.e., self-avoiding walks on n X n grid starting in top left corner.

Original entry on oeis.org

1, 2, 8, 52, 824, 22144, 1510446, 180160012, 54986690944, 29805993260994, 41433610713353366, 103271401574007978038, 660340630211753942588170, 7618229614763015717175450784, 225419381425094248494363948728158

Offset: 1

Nathaniel Johnston, Oct 03 2008

The sequence may be enumerated using standard methods for counting Hamiltonian cycles on a modified graph with two additional nodes, one joined to a corner vertex and the other joined to all other vertices. - Andrew Howroyd, Nov 08 2015

Nathaniel Johnston, On Maximal Self-Avoiding Walks
Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.

Main diagonal of A378938.
Cf. A046994, A046995, A145156, A160240, A160241.
Cf. A000532, A001184, A120443, A003763, A271507, A007764.

a(9)-a(15) from Andrew Howroyd, Nov 08 2015

A288518 Array read by antidiagonals: T(m,n) = number of (undirected) paths in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 3, 12, 3, 6, 49, 49, 6, 10, 146, 322, 146, 10, 15, 373, 1618, 1618, 373, 15, 21, 872, 7119, 14248, 7119, 872, 21, 28, 1929, 28917, 111030, 111030, 28917, 1929, 28, 36, 4118, 111360, 801756, 1530196, 801756, 111360, 4118, 36

Offset: 1

Andrew Howroyd, Jun 10 2017

Paths of length zero are not counted here.

			Table starts:
=================================================================
m\n|  1    2      3       4         5          6            7
---|-------------------------------------------------------------
1  |  0    1      3       6        10         15           21 ...
2  |  1   12     49     146       373        872         1929 ...
3  |  3   49    322    1618      7119      28917       111360 ...
4  |  6  146   1618   14248    111030     801756      5493524 ...
5  | 10  373   7119  111030   1530196   19506257    235936139 ...
6  | 15  872  28917  801756  19506257  436619868   9260866349 ...
7  | 21 1929 111360 5493524 235936139 9260866349 343715004510 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..276
Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Grid Graph

Rows 2-7 are A288516, A288527, A358800, A358801, A358802, A358803.
Main diagonal is A288032.
Cf. A231829, A064298, A271465, A271592, A120443, A001184, A145157, A003763.

A271507 Number of self-avoiding walks of any length from NW to SW corners on an n X n grid or lattice.

Original entry on oeis.org

1, 2, 11, 178, 8590, 1246850, 550254085, 741333619848, 3046540983075504, 38141694646516492843, 1453908228148524205711098, 168707605740228097581729005751, 59588304533380500951726150179910606, 64061403305026776755367065417308840021540

Offset: 1

Andrew Howroyd, Apr 08 2016

nonn

Main diagonal of A271465.

Cf. A007764, A000532, A001184, A145157, A120443, A003763.

A271465 = Cases[Import["https://oeis.org/A271465/b271465.txt", "Table"], {, }][[All, 2]];
a[n_] := A271465[[2 n^2 - 2 n + 1]];
Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Sep 23 2019 *)

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A271507(n):
    if n == 1: return 1
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n
    paths = GraphSet.paths(start, goal)
    return paths.len()
print([A271507(n) for n in range(1, 10)])  # Seiichi Manyama, Mar 21 2020

A121785 "Spanning walks" on the square lattice (see Jensen web site for further information).

Original entry on oeis.org

8, 95, 2320, 154259, 30549774, 17777600753, 30283708455564, 152480475641255213, 2287842813828061810244, 102744826737618542833764649, 13848270995235582268846758977770

Offset: 1

N. J. A. Sloane, Aug 30 2006

nonn

Number of Hamiltonian paths in the graph P_{n+1} X P_{n+1} starting at any of the n+1 vertices on one side of the graph and terminating at any of the n+1 vertices on the opposite side. - Andrew Howroyd, Apr 10 2016

Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 1..26
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 D1.
I. Jensen, Series Expansions for Self-Avoiding Walks
I. Jensen, Spanning walks (terms)

Cf. A000532, A001184, A121789, A215527.

Cf. A333323, A356610-A356616, A354511.

A333580 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in an n X k grid starting at the lower left corner and finishing in the upper right corner.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 4, 4, 1, 1, 1, 0, 8, 0, 8, 0, 1, 1, 1, 16, 20, 20, 16, 1, 1, 1, 0, 32, 0, 104, 0, 32, 0, 1, 1, 1, 64, 111, 378, 378, 111, 64, 1, 1, 1, 0, 128, 0, 1670, 0, 1670, 0, 128, 0, 1, 1, 1, 256, 624, 6706, 10204, 10204, 6706, 624, 256, 1, 1

Offset: 1

Seiichi Manyama, Mar 27 2020

			Square array T(n,k) begins:
  1, 1,  1,   1,    1,     1,      1,      1, ...
  1, 0,  1,   0,    1,     0,      1,      0, ...
  1, 1,  2,   4,    8,    16,     32,     64, ...
  1, 0,  4,   0,   20,     0,    111,      0, ...
  1, 1,  8,  20,  104,   378,   1670,   6706, ...
  1, 0, 16,   0,  378,     0,  10204,      0, ...
  1, 1, 32, 111, 1670, 10204, 111712, 851073, ...
  1, 0, 64,   0, 6706,     0, 851073,      0, ...

Andrew Howroyd, Table of n, a(n) for n = 1..378
Index entries for sequences related to graphs, Hamiltonian

Rows n=1..10 (with 0 omitted) give: A000012, A000035, A011782(n-1), A014523, A014584, A333581, A333582, A333583, A333584, A333585.
T(2*n-1,2*n-1) gives A001184(n-1).
Cf. A271592.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333580(n, k):
    if n == 1 or k == 1: return 1
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, k * n
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
print([A333580(j + 1, i - j + 1) for i in range(12) for j in range(i + 1)])

T(n,k) = T(k,n).

A272445 Numbers of paths for moving a king from a corner to the opposite one in an n X n chessboard, provided that each cell must be reached exactly once.

Original entry on oeis.org

1, 2, 30, 4942, 5853876, 58039036412, 4458135334234700, 2811002302704446996926, 14204916761279146343474398608, 580077332863329104087361516015280826, 190934226579364879405564404836420471442186330, 507088717315287736410992294230305692212344811974323748

Offset: 1

César Eliud Lozada, Apr 29 2016

			On a 2 X 2 chessboard, the king has 2 paths to go from a corner to the opposite one, standing exactly one time on each cell, so a(2)=2.
On a 3 X 3 chessboard, the king has 30 paths to go from a corner to the opposite one, standing exactly one time on each cell, so a(3)=30.

César Eliud Lozada, Chess king paths
Eric Weisstein's World of Mathematics, King Graph

Cf. A001184, A158651, A140518, A288033.

ChessKingPaths := proc(N, aUsed:=[1])
  local nLast, nPrev,  nNext,  i, j, i1, j1, i2, j2:
  global aPath, nPaths;
if N=1 then
  aPath:=[[1]]; nPaths:=1; return nPaths;
end if:
  if aUsed=[1] then
    aPath:=[]; nPaths:=0;
  end if;
  nLast:=N^(2);   #`opposite corner`
  nPrev := aUsed[-1] ; #actual position
  #Check all possible next cells
  i1:=`if`(floor((nPrev-1)/N)=0,0,-N); i2:=`if`(floor((nPrev-1)/N)=N-1,0,N);
  j1:=`if`((nPrev-1)mod N=0,0,-1); j2:=`if`((nPrev-1) mod N=N-1,0,1);
  for i from i1 to i2 by N do
    for j from j1 to j2 by 1 do
      if i=0 and j=0 then next fi; #`no move`
      nNext:=nPrev+i+j;
      #`out of bounds or already visited`
      if nNext<1 or nNext>nLast or (nNext in aUsed) then next: fi;
      #`nLast must be the last one`
      if (nNext=nLast and nops(aUsed)<>nLast-1) then next fi:
      if nNext=nLast  and nops(aUsed)=nLast-1 then  #`path completed`
        #comment if list is not required. It will consume all memory for N>=5
        aPath:=[op(aPath), [op(aUsed),nNext]];
        nPaths:=nPaths+1;
        break:
      else
        ChessKingPaths(N,[op(aUsed),nNext]) #move and continue
      end if;
    end do:
  end do:
  return nPaths;
end proc:
#For a(n) call this function with parameter n.
#Examples: ChessKingPaths(2), ChessKingPaths(3),...
#The list of paths is stored in variable aPath.

a(5)-a(11) from Andrew Howroyd, Jun 19 2017

a(12) from Ed Wynn, Jul 08 2023

A068381 Number of partitions of n X n checkerboard by two edgewise-connected sets which produce the maximum n^2-2n+2 frontier edges between the two sets.

Original entry on oeis.org

12, 32, 96, 648, 7736, 228424, 11974112, 1599762776, 382467306272, 234367651907856, 258981528765867728, 733498025032488425464, 3770347483688546402804760, 49588653272896250824990166768

Offset: 2

R. H. Hardin, Mar 04 2002

nonn

Not divided by 4 because that property may not continue.

Each partition is counted twice in this sequence. The sequence can be computed by counting Hamiltonian paths on a n-1 x n-1 grid that start at any vertex on the grid boundary and terminate at another boundary vertex. Counts for whether the path starts or terminates on a corner or non-corner need to be computed separately as there are different multiplication factors. - Andrew Howroyd, Apr 13 2016

			Illustration of a(2)=6*2:
    __.__     __.__     __.__    __.__     __.__     __.__
   |__|  |   |  |__|   |   __|  |__   |   |__.__|   |  |  |
   |__.__|   |__.__|   |__|__|  |__|__|   |__.__|   |__|__|
Illustration of relation of a Hamiltonian path in a 3 x 3 grid to solutions of a(4):
                 .__.__.__.__.   .__.__.__.__.   .__.__.__.__.   .__.__.__.__.
   .__.__        |__.__.__.  |   |  |__.__.  |   |__.__.__.  |   |  |__.__.  |
    __.__|  <=>  |  .__.__|  |   |  .__.__|  |   |  .__.__|  |   |  .__.__|  |
   |__.__.       |  |__.__.__|   |  |__.__.__|   |  |__.__.  |   |  |__.__.  |
                 |__.__.__.__|   |__.__.__.__|   |__.__.__|__|   |__.__.__|__|

Cf. A068392, A068393.

Cf. A001184, A000532, A121789.

a(7)-a(15) from Andrew Howroyd, Apr 13 2016

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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A120443 Number of (undirected) Hamiltonian paths in the n X n grid graph.

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A000532 Number of Hamiltonian paths from NW to SW corners in an n X n grid.

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A145157 Number of Greek-key tours on an n X n board; i.e., self-avoiding walks on n X n grid starting in top left corner.

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Extensions

A288518 Array read by antidiagonals: T(m,n) = number of (undirected) paths in the grid graph P_m X P_n.

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A271507 Number of self-avoiding walks of any length from NW to SW corners on an n X n grid or lattice.

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Mathematica

Python

A121785 "Spanning walks" on the square lattice (see Jensen web site for further information).

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A333580 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in an n X k grid starting at the lower left corner and finishing in the upper right corner.

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Python

Formula

A272445 Numbers of paths for moving a king from a corner to the opposite one in an n X n chessboard, provided that each cell must be reached exactly once.

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Maple