OFFSET
1,2
COMMENTS
This sequence is read in a table, thus:
m ->
1, 3, 6, 10, …
n 3, 13, 40, …
| 6, 40, …
v 10, …
…
This sequence gives the number of closed, simple loops on a rectangular lattice of dots, where the edges of the loop can be horizontal or vertical.
This is also the number of solutions to an unclued slitherlink puzzle.
Main diagonal is A140517. - Joerg Arndt, Sep 01 2014
Equivalently, the number of cycles in the grid graph P_{m+1} X P_{n+1}. - Andrew Howroyd, Jun 12 2017
LINKS
Douglas Boffey and Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..70 from Douglas Boffey)
Wikipedia, Slitherlink
EXAMPLE
Table starts:
=================================================================
m\n| 1 2 3 4 5 6 7
---|-------------------------------------------------------------
1 | 1 3 6 10 15 21 28...
2 | 3 13 40 108 275 681 1664...
3 | 6 40 213 1049 5034 23984 114069...
4 | 10 108 1049 9349 80626 692194 5948291...
5 | 15 275 5034 80626 1222363 18438929 279285399...
6 | 21 681 23984 692194 18438929 487150371 12947640143...
7 | 28 1664 114069 5948291 279285399 12947640143 603841648931...
... - Andrew Howroyd, Jun 12 2017
a(2,2) = 13, thus:
1) 2) 3) 4) 5)
+-+ + + +-+ + + + + + + +-+ +
| | | | | |
+-+ + + +-+ +-+ + + +-+ + + +
| | | | | |
+ + + + + + +-+ + + +-+ +-+ +
6) 7) 8) 9) 10)
+ +-+ +-+-+ + + + +-+ + + +-+
| | | | | | | |
+ + + +-+-+ +-+-+ + +-+ +-+ +
| | | | | | | |
+ +-+ + + + +-+-+ +-+-+ +-+-+
11) 12) 13)
+-+-+ +-+-+ +-+-+
| | | | | |
+-+ + + +-+ + + +
| | | | | |
+ +-+ +-+ + +-+-+
PROG
(Python)
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A231829(n, k):
universe = tl.grid(n, k)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
return cycles.len()
print([A231829(j + 1, i - j + 1) for i in range(9) for j in range(i + 1)]) # Seiichi Manyama, Nov 24 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Douglas Boffey, Nov 14 2013
STATUS
approved