A232103 - OEIS

%I #31 Sep 24 2019 05:06:23

%S 1,1,1,1,5,1,1,15,15,1,1,39,106,39,1,1,97,582,582,97,1,1,237,2952,

%T 6074,2952,237,1,1,575,14488,56778,56778,14488,575,1,1,1391,69982,

%U 510600,943340,510600,69982,1391,1,1,3361,335356,4502836,15009212,15009212

%N Square array read by antidiagonals: T(m,n) = number of ways of drawing a simple loop on an m x n rectangular lattice of dots in such a way that it touches each edge.

%C This sequence is to be read as a table:

%C 1,   1,   1,   1,   1, ...

%C 1,   5,  15,  39, ...

%C 1,  15, 106, ...

%C 1,  39, ...

%C 1, ...

%C ...

%C This represents the number of simple, closed loops that can be formed on an m x n lattice of dots in such a way that it touches each edge.

%C This sequence is related to A231829, called b(i,j) by a(i,j) = b(i,j) - 2 * b(i,j-1) + b(i,j-2) - 2 * b(i-1,j) + 4 * b(i-1,j-1) - 2 * b(j-1,j-2) + b(i-2,j) - 2 * b(i-2,j-1) + b(i-2,j-2).

%C Equivalently, the number of fixed polyominoes without holes that have a width of m and height of n. - _Andrew Howroyd_, Oct 04 2017

%H Andrew Howroyd, <a href="/A232103/b232103.txt">Table of n, a(n) for n = 1..325</a>

%H Jean-François Alcover, <a href="/A232103/a232103.txt">Mathematica program</a>

%F T(m, n) = U(m, n) - 2*U(m, n-1) + U(m, n-2) where U(m, n) = V(m, n) - 2*V(m-1, n) + V(m-2, n) and V(m, n) = A231829(m, n). - _Andrew Howroyd_, Oct 04 2017

%e Array begins:

%e ==============================================================

%e m\n| 1   2     3       4         5           6            7

%e ---|----------------------------------------------------------

%e 1  | 1   1     1       1         1           1            1...

%e 2  | 1   5    15      39        97         237          575...

%e 3  | 1  15   106     582      2952       14488        69982...

%e 4  | 1  39   582    6074     56778      510600      4502836...

%e 5  | 1  97  2952   56778    943340    15009212    234411981...

%e 6  | 1 237 14488  510600  15009212   419355340  11509163051...

%e 7  | 1 575 69982 4502836 234411981 11509163051 554485727288...

%e ... - _Andrew Howroyd_, Oct 04 2017

%e a(3,2) is 15, 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)       14)       15)

%e +-+-+ +   + + +-+   +-+ +-+   + +-+-+   +-+-+-+

%e |   |         | |   | | | |     |   |   |     |

%e +   +-+   +-+-+ +   + +-+ +   +-+ + +   + + + +

%e |     |   |     |   |     |   |     |   |     |

%e +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+

%Y Rows 2-3 are A034182, A293263.

%Y Main diagonal is A293261.

%Y Cf. A231829, A292357.

%K nonn,tabl

%O 1,5

%A _Douglas Boffey_, Nov 21 2013