A236679 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2)^2, read by rows.

1, 1, 1, 1, 1, 3, 4, 2, 1, 1, 3, 13, 20, 14, 1, 6, 37, 138, 277, 273, 143, 39, 7, 1, 1, 6, 75, 505, 2154, 5335, 7855, 6472, 2756, 459, 1, 10, 147, 1547, 10855, 50021, 153311, 311552, 416825, 361426, 200996, 71654, 16419, 2363, 211, 11, 1, 1, 10, 246, 3759, 39926, 291171

Offset

2,6

Comments

Changing the offset from 2 to 1, this is also the sequence: "Triangle read by rows: T(n,k) is the number of nonequivalent ways to place k non-attacking kings on an n X n board." (For if each king is represented by a 2 X 2 tile with the king in the upper left corner, the kings do not attack each other.) For example, with offset 1, T(4,3) = 20 because there are 20 nonequivalent ways to place 3 kings on a 4 X 4 chessboard so that no king threatens any other. - Heinrich Ludwig and N. J. A. Sloane, Dec 21 2016

It appears that rows 2n and 2n-1 both contain n^2 + 1 entries. Rotations and reflections of placements are not counted. If they are to be counted, see A193580. - Heinrich Ludwig, Dec 11 2016

Links

Heinrich Ludwig, Table of n, a(n) for n = 2..107

Christopher Hunt Gribble, C++ program

Formula

It appears that:

T(n,0) = 1, n>= 2

T(n,1) = (floor((n-2)/2)+1)*(floor((n-2)/2+2))/2, n >= 2

T(c+2*2,2) = A131474(c+1)*(2-1) + A000217(c+1)*floor(2^2/4) + A014409(c+2), 0 <= c < 2, c even

T(c+2*2,2) = A131474(c+1)*(2-1) + A000217(c+1)*floor((2-1)(2-3)/4) + A014409(c+2), 0 <= c < 2, c odd

T(c+2*2,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((2-c-1)/2) + A131941(c+1)*floor((2-c)/2)) + S(c+1,3c+2,3), 0 <= c < 2 where

S(c+1,3c+2,3) =

A054252(2,3), c = 0

A236679(5,3), c = 1

Example

T(4,2) = 4 because the number of equivalence classes of ways of placing 2 2 X 2 square tiles in a 4 X 4 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
._______        _______        _______        _______
| . | . |      | . |___|      | . |   |      |_______|
|___|___|      |___| . |      |___|___|      | . | . |
|       |      |   |___|      |   | . |      |___|___|
|_______|      |_______|      |___|___|      |_______|
The first 6 rows of T(n,k) are:
.\ k  0    1    2    3    4    5    6    7    8    9
n
2     1    1
3     1    1
4     1    3    4    2    1
5     1    3   13   20   14
6     1    6   37  138  277  273  143   39    7    1
7     1    6   75  505 2154 5335 7855 6472 2756  459

Crossrefs

Cf. A054252, A236560, A236757, A236800, A236829, A236865, A236915, A236936, A236939.

Row sums give A275869.

Columns 2..7: A279111, A279112, A279113, A279114, A279115, A279116.

Diagonal T(n,n) is A279117.

Cf. A193580.

Sequence in context: A019829 A200125 A091528 * A364871 A096392 A332964

Adjacent sequences: A236676 A236677 A236678 * A236680 A236681 A236682

Keyword

tabf,nonn

Author

Christopher Hunt Gribble, Jan 29 2014

Extensions

More terms from Heinrich Ludwig, Dec 11 2016 (The former entry A279118 from Heinrich Ludwig was merged into this entry by N. J. A. Sloane, Dec 21 2016)

Status

approved