A231829 Square array read by antidiagonals: T(m,n) = number of ways of creating a closed, simple loop on an m X n rectangular lattice.

1, 3, 3, 6, 13, 6, 10, 40, 40, 10, 15, 108, 213, 108, 15, 21, 275, 1049, 1049, 275, 21, 28, 681, 5034, 9349, 5034, 681, 28, 36, 1664, 23984, 80626, 80626, 23984, 1664, 36, 45, 4040, 114069, 692194, 1222363, 692194, 114069, 4040, 45

Offset: 1

Author

Douglas Boffey, Nov 14 2013

Keywords

• nonn
• tabl

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

Examples

			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)
+-+-+     +-+-+     +-+-+
|   |     |   |     |   |
+-+ +     + +-+     + + +
  | |     | |       |   |
+ +-+     +-+ +     +-+-+
		

Links

• Douglas Boffey and Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..70 from Douglas Boffey)
• Wikipedia, Slitherlink

Crossrefs

Rows 2..10 are A059020, A288637, A339117, A339118, A339119, A339120, A339121, A358707, A358785.

Main diagonal is A140517.

Cf. A288518, A003763, A222202.

Programs

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