A225803 Number T(n,k,u) of tilings of an n X k rectangle using integer-sided square tiles, reduced for symmetry, containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k < n, u >= 0, read by rows.

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 1, 1, 1, 2, 2, 1, 2, 4, 0, 2, 1, 1, 4, 13, 10, 6, 3, 1, 0, 0, 1, 1, 1, 3, 4, 1, 1, 3, 8, 3, 2, 3, 0, 0, 1, 1, 6, 23, 33, 24, 15, 6, 0, 2, 2, 2, 1, 1, 6, 40, 101, 129, 79, 74, 53, 13, 9, 11, 4, 0, 0, 0, 0, 1

Offset

1,7

Comments

The number of entries per row is given by A225568(n>0 and n != A000217(1:)).

Links

Christopher Hunt Gribble, Rows 1..28 for n = 2..8 and k = 1..n-1 flattened

Christopher Hunt Gribble, C++ program

Formula

T1(n,k,0) = 1, T1(n,k,1) = floor(n/2)*floor(k/2).

Example

The irregular triangle T(n,k,u) begins:
n,k\u  0   1   2   3   4   5   6   7   8   9  10  11  12 ...
2,1    1
3,1    1
3,2    1   1
4,1    1
4,2    1   2   1
4,3    1   2   2   0   1
5,1    1
5,2    1   2   2
5,3    1   2   4   0   2   1
5,4    1   4  13  10   6   3   1   0   0   1
6,1    1
6,2    1   3   4   1
6,3    1   3   8   3   2   3   0   0   1
6,4    1   6  23  33  24  15   6   0   2   2   1
6,5    1   6  40 101  79  74  53  13   9  11   4   0   0 ...
...
T(5,3,2) = 4 because there are 4 different sets of tilings of the 5 X 3 rectangle by integer-sided squares in which each tiling contains 2 isolated nodes.  Any sequence of group D2 operations will transform each tiling in a set into another in the same set.  Group D2 operations are:
.   the identity operation
.   rotation by 180 degrees
.   reflection about a horizontal axis through the center
.   reflection about a vertical axis through the center
A 2 X 2 square contains 1 isolated node.  Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors.  An example of a tiling in each set is:
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 0 1 1    1 0 1 1    1 0 1 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 1 0 1    1 1 1 1    1 1 1 1
   1 1 1 1    1 1 1 1    1 0 1 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1

Crossrefs

Cf. A224239, A227004, A227690, A224850, A224861, A224867, A225777, A225542.

Sequence in context: A169676 A355912 A034798 * A357458 A349277 A307014

Adjacent sequences: A225800 A225801 A225802 * A225804 A225805 A225806

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 28 2013

Status

approved