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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.

%I A231829 #58 Nov 30 2022 11:20:27
%S A231829 1,3,3,6,13,6,10,40,40,10,15,108,213,108,15,21,275,1049,1049,275,21,
%T A231829 28,681,5034,9349,5034,681,28,36,1664,23984,80626,80626,23984,1664,36,
%U A231829 45,4040,114069,692194,1222363,692194,114069,4040,45
%N 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.
%C A231829 This sequence is read in a table, thus:
%C A231829     m ->
%C A231829     1,  3,  6, 10, …
%C A231829 n   3, 13, 40, …
%C A231829 |   6, 40, …
%C A231829 v  10, …
%C A231829     …
%C A231829 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.
%C A231829 This is also the number of solutions to an unclued slitherlink puzzle.
%C A231829 Main diagonal is A140517. - _Joerg Arndt_, Sep 01 2014
%C A231829 Equivalently, the number of cycles in the grid graph P_{m+1} X P_{n+1}. - _Andrew Howroyd_, Jun 12 2017
%H A231829 Douglas Boffey and Andrew Howroyd, <a href="/A231829/b231829.txt">Table of n, a(n) for n = 1..325</a> (terms 1..70 from Douglas Boffey)
%H A231829 Wikipedia, <a href="https://en.wikipedia.org/wiki/Slitherlink">Slitherlink</a>
%e A231829 Table starts:
%e A231829 =================================================================
%e A231829 m\n|  1    2      3       4         5           6            7
%e A231829 ---|-------------------------------------------------------------
%e A231829 1  |  1    3      6      10        15          21           28...
%e A231829 2  |  3   13     40     108       275         681         1664...
%e A231829 3  |  6   40    213    1049      5034       23984       114069...
%e A231829 4  | 10  108   1049    9349     80626      692194      5948291...
%e A231829 5  | 15  275   5034   80626   1222363    18438929    279285399...
%e A231829 6  | 21  681  23984  692194  18438929   487150371  12947640143...
%e A231829 7  | 28 1664 114069 5948291 279285399 12947640143 603841648931...
%e A231829 ... - _Andrew Howroyd_, Jun 12 2017
%e A231829 a(2,2) = 13, thus:
%e A231829 1)        2)        3)        4)        5)
%e A231829 +-+ +     + +-+     + + +     + + +     +-+ +
%e A231829 | |         | |                         | |
%e A231829 +-+ +     + +-+     +-+ +     + +-+     + + +
%e A231829                     | |         | |     | |
%e A231829 + + +     + + +     +-+ +     + +-+     +-+ +
%e A231829 6)        7)        8)        9)        10)
%e A231829 + +-+     +-+-+     + + +     +-+ +     + +-+
%e A231829   | |     |   |               | |         | |
%e A231829 + + +     +-+-+     +-+-+     + +-+     +-+ +
%e A231829   | |               |   |     |   |     |   |
%e A231829 + +-+     + + +     +-+-+     +-+-+     +-+-+
%e A231829 11)       12)       13)
%e A231829 +-+-+     +-+-+     +-+-+
%e A231829 |   |     |   |     |   |
%e A231829 +-+ +     + +-+     + + +
%e A231829   | |     | |       |   |
%e A231829 + +-+     +-+ +     +-+-+
%o A231829 (Python)
%o A231829 # Using graphillion
%o A231829 from graphillion import GraphSet
%o A231829 import graphillion.tutorial as tl
%o A231829 def A231829(n, k):
%o A231829     universe = tl.grid(n, k)
%o A231829     GraphSet.set_universe(universe)
%o A231829     cycles = GraphSet.cycles()
%o A231829     return cycles.len()
%o A231829 print([A231829(j + 1, i - j + 1) for i in range(9) for j in range(i + 1)])  # _Seiichi Manyama_, Nov 24 2020
%Y A231829 Rows 2..10 are A059020, A288637, A339117, A339118, A339119, A339120, A339121, A358707, A358785.
%Y A231829 Main diagonal is A140517.
%Y A231829 Cf. A288518, A003763, A222202.
%K A231829 nonn,tabl
%O A231829 1,2
%A A231829 _Douglas Boffey_, Nov 14 2013