A227004 Irregular triangle read by rows: T(n,k) is the number of inequivalent tilings by squares of an n X n square lattice that contain k nodes unconnected to any of their neighbors.

1, 1, 1, 1, 1, 0, 0, 1, 1, 3, 4, 2, 2, 0, 0, 0, 0, 1, 1, 3, 13, 20, 17, 6, 10, 5, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 6, 37, 138, 280, 300, 255, 218, 98, 43, 55, 28, 20, 11, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1,10

Comments

The n-th row contains (n-1)^2 + 1 elements.

The irregular triangle is shown below.

\ k 0 1 2 3 4 5 6 7 8 9 ...

n

1 1

2 1 1

3 1 1 0 0 1

4 1 3 4 2 2 0 0 0 0 1

5 1 3 13 20 17 6 10 5 0 1 ...

6 1 6 37 138 280 300 255 218 98 43 ...

7 1 6 75 505 2160 5410 8508 9179 8805 7917 ...

Links

Christopher Hunt Gribble, Table of n, a(n) for n = 1..98

Christopher Hunt Gribble, C++ program for A226978, A226979, A226980, A226981, A227004

Formula

Sum_{k=0..(n-1)^2} T(n,k) = A224239(n).

Example

For n = 4, there are 3 inequivalent tilings that contain 1 isolated node, so T(4,1) = 3.
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 3 tilings are:
1 1 1 1 1    1 1 1 1 1    1 1 1 1 1
1 0 1 1 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 1 1 1    1 1 1 1 1    1 1 1 1 1
1 1 1 1 1    1 1 1 1 1    1 1 1 1 1

Crossrefs

Cf. A224239.

Sequence in context: A326765 A096411 A228340 * A205786 A213812 A143486

Adjacent sequences: A227001 A227002 A227003 * A227005 A227006 A227007

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jun 26 2013

Status

approved