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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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

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 ...

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

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: A093324 A169676 A034798 * A307014 A240871 A236765

Adjacent sequences:  A225800 A225801 A225802 * A225804 A225805 A225806

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Jul 28 2013

STATUS

approved

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Last modified August 14 10:47 EDT 2020. Contains 336480 sequences. (Running on oeis4.)