OFFSET
0,3
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
MATHEMATICA
Table[Length[FindCycle[GridGraph[{n, n}], Infinity, All]], {n, 6}] (* Eric W. Weisstein, Mar 07 2023 *)
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
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