OFFSET
1,4
COMMENTS
Skew-tetrominoes are tiles of the form:
___
|_ |_
|___|
together with all rotations/reflections of this shape.
It is not hard to see that skew-tetrominoes cannot completely tile an n X n square, so a(n) < n^2/4.
The odd terms are easily understood: a(2m+1) = m^2.
A straightforward (greedy) construction shows that m^2 skew-tetrominoes (all with the same orientation) can be packed into a (2m+1) X (2m) rectangle. Therefore a(2m+1) >= m^2.
On the other hand, we also have a(2m+1) <= m^2: Consider all cells with indices of the form (2i, 2j); there are m^2 such cells in a (2m+1) X (2m+1) square. Moreover, any valid placement of a skew-tetromino must cover one of these cells, so a(2m+1) <= m^2.
The behavior of a(2m) appears more subtle; the initial terms satisfy a(2m) = m^2 - floor(m/2), but this formula breaks down at a(20) = 96 (not 95).
Additional terms:
(Lower bounds are from explicit constructions; upper bounds are from mixed-integer-programming search.)
a(22) in {116, 117}.
a(23) = 121.
a(24) = 139.
a(25) = 144.
a(26) in {163, 164}.
a(27) = 169.
a(28) in {190, 191}.
a(29) = 196.
a(30) in {218, 219, 220}.
a(31) = 225.
a(32) in {248, 249, 250, 251}.
a(33) = 256.
a(34) in {280, 281, 282, 283}.
a(35) = 289.
a(36) in {316, 317, 318}.
a(37) = 324.
a(38) in {352, 353, 354, 355}.
a(39) = 361.
a(40) in {388, 389, 390, 391, 392, 393, 394}.
a(41) = 400.
a(42) in {432, 433, 434}.
LINKS
Alexander D. Healy, Examples of (near-)optimal packings for n <= 42
Wikipedia, Tetromino
FORMULA
a(n) < n^2/4.
a(2m+1) = m^2.
EXAMPLE
a(8) = 14 by the following packing of 14 skew-tetrominoes into an 8 X 8 square:
_______________
|_|1 _| |___| |_|
|___| 2_|3 _|_4 |
|_ 5|_|___| | |_|
| |___| | 6_|_7 |
|_8 | 9_|_|_10|_|
| |_|_|11_| |___|
|_12|___|13_|14_|
|_|_|___|_|___|_|
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Alexander D. Healy, Jul 03 2022
STATUS
approved