OFFSET
1,5
COMMENTS
This sequence is to be read as a table:
1, 1, 1, 1, 1, ...
1, 5, 15, 39, ...
1, 15, 106, ...
1, 39, ...
1, ...
...
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.
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).
Equivalently, the number of fixed polyominoes without holes that have a width of m and height of n. - Andrew Howroyd, Oct 04 2017
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..325
Jean-François Alcover, Mathematica program
FORMULA
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
EXAMPLE
Array begins:
==============================================================
m\n| 1 2 3 4 5 6 7
---|----------------------------------------------------------
1 | 1 1 1 1 1 1 1...
2 | 1 5 15 39 97 237 575...
3 | 1 15 106 582 2952 14488 69982...
4 | 1 39 582 6074 56778 510600 4502836...
5 | 1 97 2952 56778 943340 15009212 234411981...
6 | 1 237 14488 510600 15009212 419355340 11509163051...
7 | 1 575 69982 4502836 234411981 11509163051 554485727288...
... - Andrew Howroyd, Oct 04 2017
a(3,2) is 15, thus:
1) 2) 3) 4) 5)
+-+-+-+ +-+-+-+ + +-+-+ +-+-+-+ +-+-+-+
| | | | | | | | | |
+ +-+-+ +-+ +-+ +-+ +-+ + + +-+ +-+-+ +
| | | | | | | | | |
+-+ + + + +-+ + +-+-+ + +-+-+ + + + +-+
6) 7) 8) 9) 10)
+-+-+-+ +-+-+ + +-+-+-+ +-+ + + + +-+ +
| | | | | | | | | |
+ +-+ + +-+ +-+ +-+ + + + +-+-+ +-+ +-+
| | | | | | | | | | | |
+-+ +-+ + +-+-+ + +-+-+ +-+-+-+ +-+-+-+
11) 12) 13) 14) 15)
+-+-+ + + + +-+ +-+ +-+ + +-+-+ +-+-+-+
| | | | | | | | | | | |
+ +-+ +-+-+ + + +-+ + +-+ + + + + + +
| | | | | | | | | |
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Douglas Boffey, Nov 21 2013
STATUS
approved