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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
OFFSET
1,7
The number of entries per row is given by A225568(n>0 and n != A000217(1:)).
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
Sequence in context: A169676 A355912 A034798 * A357458 A349277 A307014
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved

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Last modified September 20 06:01 EDT 2024. Contains 376024 sequences. (Running on oeis4.)