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