OFFSET

1,7

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

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Jul 28 2013

STATUS

approved