A227690
Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles reduced for symmetry; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 6, 6, 5, 1, 1, 1, 1, 9, 10, 13, 10, 9, 1, 1, 1, 1, 12, 21, 39, 39, 21, 12, 1, 1, 1, 1, 21, 39, 115, 77, 115, 39, 21, 1, 1, 1, 1, 30, 82, 295, 521, 521, 295, 82, 30, 1, 1
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 2, 4, 5, 9, 12, 21, ...
1, 1, 2, 3, 6, 10, 21, 39, 82, ...
1, 1, 4, 6, 13, 39, 115, 295, 861, ...
1, 1, 5, 10, 39, 77, 521, 1985, 8038, ...
1, 1, 9, 21, 115, 521, 1494, 15129, 83609, ...
1, 1, 12, 39, 295, 1985, 15129, 56978, 861159, ...
1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023, ...
...
A(4,3) = 6 because there are 6 ways to tile a 3 X 4 rectangle by subsquares, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct:
._____ _. ._______. ._______.
| |_| | | | | |_|_|
| |_| |___|_ _| |___| |
|_____|_| |_|_|_|_| |_|_|___|
._______. ._______. ._______.
| |_|_| |_| |_| |_|_|_|_|
|___|_|_| |_|___|_| |_|_|_|_|
|_|_|_|_| |_|_|_|_| |_|_|_|_|
A359020
Number of inequivalent tilings of a 4 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 4, 6, 13, 39, 115, 295, 861, 2403, 7048, 20377, 60008, 175978, 519589, 1532455, 4531277, 13395656, 39639758, 117301153, 347248981, 1028011708, 3043852214, 9012879842, 26689014028, 79033362580, 234045889421, 693101137571, 2052569508948
Offset: 0
a(3) is 6 because of:
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
| | | | | | | | | | | | | | | | | | |
+-+-+-+ + + + +-+ + +-+ + +-+ +-+-+-+
| | | | | | | | | | | | | | | | | |
+-+-+-+ + + +-+-+-+ +-+-+-+ +-+-+-+ + +-+
| | | | | | | | | | | | | | | | | | |
+-+-+-+ +-+-+-+ + +-+ +-+ + +-+-+-+ +-+-+-+
| | | | | | | | | | | | | | | | | | | |
+-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A359021
Number of inequivalent tilings of a 5 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 5, 10, 39, 77, 521, 1985, 8038, 32097, 130125, 525676, 2131557, 8635656, 35017970, 141968455, 575692056, 2334344849, 9465939422, 38384559168, 155652202456, 631178976378, 2559476952229, 10378857744374, 42087027204278, 170665938023137, 692062856184512
Offset: 0
a(2) is 5 because of:
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
| | | | | | | | | | |
+-+-+ +-+-+ + + + + +-+-+
| | | | | | | | | | |
+-+-+ + + +-+-+ +-+-+ + +
| | | | | | | | | | | |
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
| | | | | | | | | | | | |
+-+-+ + + + + +-+-+ +-+-+
| | | | | | | | | | | | |
+-+-+ +-+-+ +-+-+ +-+-+ +-+-+
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A359022
Number of inequivalent tilings of a 6 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 9, 21, 115, 521, 1494, 15129, 83609, 459957, 2551794, 14150081, 78597739
Offset: 0
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A359023
Number of inequivalent tilings of a 7 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 12, 39, 295, 1985, 15129, 56978, 861159, 6542578, 49828415
Offset: 0
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A359024
Number of inequivalent tilings of an 8 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 21, 82, 861, 8038, 83609, 861159, 4495023
Offset: 0
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A359025
Number of inequivalent tilings of a 9 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 30, 163, 2403, 32097, 459957, 6542578, 93604244
Offset: 0
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A359026
Number of inequivalent tilings of a 10 X n rectangle using integer-sided square tiles.
Original entry on oeis.org
1, 1, 51, 347, 7048, 130125, 2551794, 49828415
Offset: 0
Sequences for fixed and free (inequivalent) tilings of m X n rectangles, for 2 <= m <= 10:
A362261
Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle, up to rotations and reflections.
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 8, 12, 22, 40, 73, 146, 292, 560, 1120, 2532, 5040, 10080, 22176, 44352, 88704, 192272, 384384, 768768, 1647360, 3294720, 6589440, 14003120, 28006240, 56012480, 126028080, 266053680, 532107360, 1182438400, 2483130720, 4966261440, 10925775168
Offset: 0
-
from math import comb
def F(i,j,k):
# total number of tilings using i, j, and 2*j+3*k squares of side lengths 3, 2, and 1, respectively
return comb(i+j+k,i)*comb(j+k,j)*2**j
def F0(i,j,k):
# number of inequivalent tilings
x = F(i,j,k)
if j == 0: x += comb(i+k,i) # horizontal line of symmetry
if i%2+j%2+k%2 <= 1: x += 2*F(i//2,j//2,k//2) # vertical line of symmetry or rotational symmetry
return x//4
def A362261(n):
return max(F0(i,j,n-3*i-2*j) for i in range(n//3+1) for j in range((n-3*i)//2+1))
A361525
Number of ways of dividing an n X 3 rectangle into integer-sided rectangles, up to rotations and reflections.
Original entry on oeis.org
1, 3, 17, 54, 892, 8159, 80021, 791821, 7906439, 79069308
Offset: 0
Showing 1-10 of 10 results.