A140517 Number of cycles in an n X n grid.

0, 1, 13, 213, 9349, 1222363, 487150371, 603841648931, 2318527339461265, 27359264067916806101, 988808811046283595068099, 109331355810135629946698361371, 36954917962039884953387868334644457, 38157703688577845304445530851242055267353, 120285789340533859558405124213592877516931371715

Offset: 0

Author

Don Knuth, Jul 26 2008

Keywords

• nonn

Comments

Or, number of simply connected and rookwise connected regions of an (n-1) X (n-1) grid.

References

• D. E. Knuth, The Art of Computer Programming, Volume 4A, Section 7.1.4.

Links

• Artem M. Karavaev and Hiroaki Iwashita, Table of n, a(n) for n = 0..26 (A. Karavaev computed terms 10 to 19)
• Fawaz Alazemi, Arash Azizimazreah, Bella Bose, and Lizhong Chen, Routerless Networks-on-Chip, U.S. Patent Application No. 15,445,736, 2017.
• 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, Division of Computer Science, Hokkaido University, Report Series A, April 26 2013.
• Artem M. Karavaev, FlowProblem.Ru: All Simple Cycles
• Kimberly Villalobos, Vilim Štih, Amineh Ahmadinejad, Shobhita Sundaram, Jamell Dozier, Andrew Francl, Frederico Azevedo, Tomotake Sasaki, and Xavier Boix, Do Neural Networks for Segmentation Understand Insideness?, MIT Center for Brains, Minds + Machines, CBMM Memo (2020) No. 105.
• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Grid Graph
• Wikipedia, ZDD

Crossrefs

Corner-to-corner paths in this grid are enumerated in A007764.

Main diagonal of A231829.

Programs

Mathematica

Table[Length[FindCycle[GridGraph[{n, n}], Infinity, All]], {n, 6}] (* Eric W. Weisstein, Mar 07 2023 *)

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A140517(n):
    if n == 0: return 0
    universe = tl.grid(n, n)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    return cycles.len()
print([A140517(n) for n in range(9)])  # Seiichi Manyama, Mar 23 2020

Extensions

a(9) calculated using the ZDD technique described in Knuth's The Art of Computer Programming, Exercises 7.1.4, by Ashutosh Mehra, Dec 19 2008

a(10)-a(19) calculated using a transfer matrix method by Artem M. Karavaev, Jun 03 2009, Oct 20 2009

a(20)-a(26) calculated by Hiroaki Iwashita, Apr 26 2013, Nov 18 2013

Edited by Max Alekseyev, Jan 24 2018