OFFSET
1,2
COMMENTS
A219158 gives the minimum number of squares to tile an i x j rectangle. a(n) is found by checking all rectangles (i,j) for which A219158 has a dissection into n squares.
Due to the potential counterexamples to the minimal squaring conjecture (see MathOverflow link), terms after a(19) have to be considered only as lower bounds: a(20) >= 876696755, a(21) >= 2735106696. - Hugo Pfoertner, Nov 17 2020, Apr 02 2021
LINKS
Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles.
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.
Bertram Felgenhauer, Filling rectangles with integer-sided squares.
MathOverflow, tiling a rectangle with the smallest number of squares, answer by Ed Pegg Jr, Jul 09 2017.
Rainer Rosenthal, Rectangle tiled by 19 squares with maximum area a(19)
EXAMPLE
a(6) = 11*13 = 143.
Dissection of the 11 X 13 rectangle into 6 squares:
.
+-----------+-------------+
| | |
| | |
| 6 X 6 | 7 X 7 |
| | |
| | |
+---------+-+ |
| +-+-----+-------+
| 5 X 5 | | |
| | 4 X 4 | 4 X 4 |
| | | |
+---------+-------+-------+
.
a(19) = 16976*17061 = 289627536.
Dissection of the 16976 X 17061 rectangle into 19 squares:
.
+----------------+-------------+
| | |
| | |
| | 7849 |
| 9212 | |
| | |
| | |
| |------+------|
|________________| | |
| | see | 4109 |
| |Rosenthal| |
| | link +-+------+
| 7764 |-------| |
| | | 5018 |
| | 4279 | |
| | | |
+-------------+-------+--------+
.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Rainer Rosenthal, Nov 12 2020
EXTENSIONS
a(11)-a(17) from Hugo Pfoertner based on data from squaring.net website, Nov 17 2020
a(18) from Hugo Pfoertner, Feb 18 2021
a(19) from Hugo Pfoertner, Apr 02 2021
STATUS
approved