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A225777 Number T(n,k,u) of distinct tilings of an n X k rectangle using integer-sided square tiles 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. 5
1, 1, 1, 1, 1, 1, 2, 1, 4, 0, 0, 1, 1, 1, 3, 1, 1, 6, 4, 0, 2, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 1, 4, 3, 1, 8, 12, 0, 3, 4, 1, 12, 37, 34, 15, 12, 4, 0, 0, 2, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 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.

LINKS

Christopher Hunt Gribble, Rows 1..36 for n=2..8 and k=1..n flattened

Christopher Hunt Gribble, C++ program

FORMULA

T(n,k,0) = 1, T(n,k,1) = (n-1)(k-1), T(n,k,2) = (n^2(k-1) - n(2k^2+5k-13) + (k^2+13k-24))/2.

Sum_{u=1..(n-1)^2} T(n,n,u) = A045846(n).

EXAMPLE

The irregular triangle begins:

n,k\u 0   1   2   3   4   5   6   7   8   9  10  11  12 ...

1,1   1

2,1   1

2,2   1   1

3,1   1

3,2   1   2

3,3   1   4   0   0   1

4,1   1

4,2   1   3   1

4,3   1   6   4   0   2

4,4   1   9  16   8   5   0   0   0   0   1

5,1   1

5,2   1   4   3

5,3   1   8  12   0   3   4

5,4   1  12  37  34  15  12   4   0   0   2

5,5   1  16  78 140  88  44  68  32   0   4   0   0   0 ...

...

For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. 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.  Then the 4 tilings are:

   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 1 0 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 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. A045846, A219924, A226997, A225542, A225803, A225568.

Sequence in context: A096097 A212805 A246431 * A016585 A309054 A143316

Adjacent sequences:  A225774 A225775 A225776 * A225778 A225779 A225780

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Jul 26 2013

STATUS

approved

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Last modified September 28 04:51 EDT 2021. Contains 347703 sequences. (Running on oeis4.)