

A140517


Number of cycles in an n X n grid.


16



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

0,3


COMMENTS

Or, number of simply connected and rookwise connected regions of an (n1) X (n1) 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, Lizhong Chen, Routerless NetworksonChip, 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 TRA1364, 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, 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


PROG

(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


CROSSREFS

Cornertocorner paths in this grid are enumerated in A007764.
Main diagonal of A231829.
Sequence in context: A251093 A132542 A069989 * A096141 A218475 A294982
Adjacent sequences: A140514 A140515 A140516 * A140518 A140519 A140520


KEYWORD

nonn


AUTHOR

Don Knuth, Jul 26 2008


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


STATUS

approved



