A064297 -id:A064297 - 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

A064298 Square array read by antidiagonals of self-avoiding rook paths joining opposite corners of n X k board.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 8, 1, 1, 16, 38, 38, 16, 1, 1, 32, 125, 184, 125, 32, 1, 1, 64, 414, 976, 976, 414, 64, 1, 1, 128, 1369, 5382, 8512, 5382, 1369, 128, 1, 1, 256, 4522, 29739, 79384, 79384, 29739, 4522, 256, 1, 1, 512, 14934, 163496, 752061, 1262816, 752061, 163496, 14934, 512, 1

Offset: 1

Henry Bottomley, Sep 05 2001

			The start of the sequence as table:
* 1  1    1     1      1        1         1 ...
* 1  2    4     8     16       32        64 ...
* 1  4   12    38    125      414      1369 ...
* 1  8   38   184    976     5382     29739 ...
* 1 16  125   976   8512    79384    752061 ...
* 1 32  414  5382  79384  1262816  20562673 ...
* 1 64 1369 29739 752061 20562673 575780564 ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 331-339.

Andrew Howroyd, Table of n, a(n) for n = 1..378
Steven R. Finch, Self-Avoiding Walks of a Rook on a Chessboard [From Steven Finch, Apr 20 2019]
Steven R. Finch, Self-Avoiding Walks of a Rook [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]
Steven R. Finch, Table of Non-Overlapping Rook Paths [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]

A064297 together with its transpose.
Rows and columns include A000012, A000079, A006192, A007786, A007787, A145403, A333812.
Main diagonal is A007764.
Cf. A271465.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A064298(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)
    return paths.len()
print([A064298(j + 1, i - j + 1) for i in range(11) for j in range(i + 1)])  # Seiichi Manyama, Apr 06 2020

A006192 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of 3 X n board.

Original entry on oeis.org

1, 4, 12, 38, 125, 414, 1369, 4522, 14934, 49322, 162899, 538020, 1776961, 5868904, 19383672, 64019918, 211443425, 698350194, 2306494009, 7617832222, 25159990674, 83097804242, 274453403399, 906458014440

Offset: 1

N. J. A. Sloane

H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 331-339.
Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..150
H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178. (Annotated scanned copy)
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Steven R. Finch, Self-Avoiding Walks of a Rook on a Chessboard [From Steven Finch, Apr 20 2019]
Steven R. Finch, Self-Avoiding Walks of a Rook [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]
Steven R. Finch, Table of Non-Overlapping Rook Paths [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]
D. G. Radcliffe, N. J. A. Sloane, C. Cole, J. Gillogly, & D. Dodson, Emails, 1994
Index entries for linear recurrences with constant coefficients, signature (4,-3,2,1).

Cf. A064297, A064298, A007786, A007787, A007764.

I:=[1,4,12,38]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Oct 06 2011

LinearRecurrence[{4,-3,2,1},{1,4,12,38},40] (* Harvey P. Dale, Oct 05 2011 *)

a(n) = 4*a(n-1) - 3*a(n-2) + 2*a(n-3) + a(n-4) with a(0) = 0, a(1) = 1, a(2) = 4 and a(3) = 12. - Henry Bottomley, Sep 05 2001

G.f.: x*(1-x^2)/(1 - 4*x + 3*x^2 - 2*x^3 - x^4). - Emeric Deutsch, Dec 22 2004

A351108 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 2, 3, 8, 3, 3, 7, 12, 40, 5, 7, 13, 26, 160, 1380, 9, 13, 28, 61, 918, 12940, 211164, 14, 27, 61, 161, 7260, 142453, 4997155, 205331148

Offset: 1

Pontus von Brömssen, Feb 01 2022

			Triangle begins:
  m\n|  1  2  3   4    5      6       7         8
  ---+-------------------------------------------
  1  |  1
  2  |  1  0
  3  |  1  1  2
  4  |  2  2  3   8
  5  |  3  3  7  12   40
  6  |  5  7 13  26  160   1380
  7  |  9 13 28  61  918  12940  211164
  8  | 14 27 61 161 7260 142453 4997155 205331148

Wikipedia, Racetrack

Cf. A064297, A291896 (column n=1), A351042, A351106, A351109 (main diagonal).

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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A064298 Square array read by antidiagonals of self-avoiding rook paths joining opposite corners of n X k board.

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A006192 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of 3 X n board.

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A351108 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using von Neumann neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

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