A379625
Triangle read by rows: T(n,k) is the number of free polyominoes with n cells whose difference between length and width is k, n >= 1, k >= 0.
Original entry on oeis.org
1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 6, 2, 3, 0, 1, 7, 16, 6, 5, 0, 1, 25, 39, 27, 11, 5, 0, 1, 80, 120, 97, 45, 19, 7, 0, 1, 255, 425, 307, 191, 71, 28, 7, 0, 1, 795, 1565, 1077, 706, 347, 115, 40, 9, 0, 1, 2919, 5217, 4170, 2505, 1454, 574, 171, 53, 9, 0, 1, 10378, 18511, 15164, 10069, 5481, 2740, 919, 257, 69, 11, 0, 1
Offset: 1
Triangle begins:
1;
0, 1;
1, 0, 1;
1, 3, 0, 1;
6, 2, 3, 0, 1;
7, 16, 6, 5, 0, 1;
25, 39, 27, 11, 5, 0, 1;
80, 120, 97, 45, 19, 7, 0, 1;
255, 425, 307, 191, 71, 28, 7, 0, 1;
795, 1565, 1077, 706, 347, 115, 40, 9, 0, 1;
2919, 5217, 4170, 2505, 1454, 574, 171, 53, 9, 0, 1;
10378, 18511, 15164, 10069, 5481, 2740, 919, 257, 69, 11, 0, 1;
...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 0 there are six free pentominoes with length 3 and width 3 as shown below, thus the difference between length and width is 3 - 3 = 0, so T(5,0) = 6.
_ _ _ _ _ _ _ _ _ _
_|_|_| |_|_|_| |_| |_|_ _|_|_ |_|_|
|_|_| |_| |_|_ _ |_|_|_ |_|_|_| |_|_
|_| |_| |_|_|_| |_|_| |_| |_|_|
.
For k = 1 there are two free pentominoes with length 3 and width 2 as shown below, thus the difference between length and width is 3 - 2 = 1, so T(5,1) = 2.
_ _ _ _
|_|_| |_|_|
|_|_| |_|_
|_| |_|_|
.
For k = 2 there are three free pentominoes with length 4 and width 2 as shown below, thus the difference between length and width is 4 - 2 = 2, so T(5,2) = 3.
_ _ _
|_| _|_| _|_|
|_| |_|_| |_|_|
|_|_ |_| |_|
|_|_| |_| |_|
.
For k = 3 there are no free pentominoes whose difference between length and width is 3, so T(5,3) = 0.
For k = 4 there is only one free pentomino with length 5 and width 1 as shown below, thus the difference between length and width is 5 - 1 = 4, so T(5,4) = 1.
_
|_|
|_|
|_|
|_|
|_|
.
Therefore the 5th row of the triangle is [6, 2, 3, 0, 1] and the row sum is A000105(5) = 12.
Note that for n = 6 and k = 1 there are 15 free polyominoes with length 4 and width 3 thus the difference between length and width is 4 - 3 = 1. Also there is a free polyomino with length 3 and width 2 thus the difference between length and width is 3 - 2 = 1, so T(6,1) = 15 + 1 = 16.
.
Row sums except the column 1 give
A259087.
Cf.
A109613,
A379623,
A379624,
A379626,
A379627,
A379628,
A379629,
A379637,
A379638,
A380283,
A380284.
A379637
Irregular triangle read by rows: T(n,k) is the sum of the widths of the free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).
Original entry on oeis.org
1, 1, 1, 2, 1, 8, 1, 10, 18, 1, 24, 66, 1, 36, 213, 72, 1, 74, 579, 552, 1, 120, 1470, 2644, 365, 1, 234, 3663, 10188, 3845, 1, 400, 9033, 33668, 25945, 1530, 1, 758, 22179, 104656, 129600, 22458, 1, 1338, 54075, 312296, 544170, 192228, 6650, 1, 2500, 131541, 919524, 2041085, 1211736, 117733
Offset: 1
Triangle begins:
1;
1;
1, 2;
1, 8;
1, 10, 18;
1, 24, 66;
1, 36, 213, 72;
1, 74, 579, 552;
1, 120, 1470, 2644, 365;
1, 234, 3663, 10188, 3845;
1, 400, 9033, 33668, 25945, 1530;
1, 758, 22179, 104656, 129600, 22458;
1, 1338, 54075, 312296, 544170, 192228, 6650;
1, 2500, 131541, 919524, 2041085, 1211736, 117733;
...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 1 there is only one free pentomino of width 1 as shown below, so T(5,1) = 1.
_
|_|
|_|
|_|
|_|
|_|
.
For k = 2 there are five free pentominoes of width 2 as shown below, hence the sum of the widths is 2 + 2 + 2 + 2 + 2 = 5*2 = 10, so T(5,2) = 10.
_ _ _
|_| _|_| _|_| _ _ _ _
|_| |_|_| |_|_| |_|_| |_|_|
|_|_ |_| |_| |_|_| |_|_
|_|_| |_| |_| |_| |_|_|
.
For k = 3 there are six free pentominoes of width 3 as shown below, hence the sum of the widths is 3 + 3 + 3 + 3 + 3 + 3 = 6*3 = 18, so T(5,3) = 18.
_ _ _ _ _ _ _ _ _ _
_|_|_| |_|_|_| |_| |_|_ _|_|_ |_|_|
|_|_| |_| |_|_ _ |_|_|_ |_|_|_| |_|_
|_| |_| |_|_|_| |_|_| |_| |_|_|
.
Therefore the 5th row of the triangle is [1, 10, 18].
.
A380283
Irregular triangle read by rows: T(n,k) is the number of regions between the free polyominoes, with n cells and width k, and their bounding boxes, n >= 1, 1 <= k <= ceiling(n/2).
Original entry on oeis.org
0, 0, 0, 1, 0, 5, 0, 7, 14, 0, 19, 52, 0, 34, 173, 48, 0, 74, 503, 384, 0, 134, 1368, 1918, 210, 0, 282, 3642, 7742, 2307, 0, 524, 9552, 26843, 16267, 752, 0, 1064, 24889, 87343, 84789, 11556, 0, 2017, 64200, 272599, 370799, 103336, 2833, 0, 4009, 164826, 838160, 1445347, 678863, 52437
Offset: 1
Triangle begins:
0;
0;
0, 1;
0, 5;
0, 7, 14;
0, 19, 52;
0, 34, 173, 48;
0, 74, 503, 384;
0, 134, 1368, 1918, 210;
0, 282, 3642, 7742, 2307;
0, 524, 9552, 26843, 16267, 752;
0, 1064, 24889, 87343, 84789, 11556;
0, 2017, 64200, 272599, 370799, 103336, 2833;
0, 4009, 164826, 838160, 1445347, 678863, 52437;
0, 7663, 420373, 2539843, 5240853, 3659815, 560348, 10396;
0, 15031, 1068181, 7631249, 18171771, 17199831, 4373770, 226716;
...
Illustration for n = 5:
The free polyominoes with five cells are also called free pentominoes.
For k = 1 there is only one free pentomino of width 1 as shown below, and there are no regions between the pentomino and its bounding box, so T(5,1) = 0.
_
|_|
|_|
|_|
|_|
|_|
.
For k = 2 there are five free pentominoes of width 2 as shown below, and from left to right there are respectively 1, 2, 2, 1, 1 regions between the pentominoes and their bounding boxes, hence the total number of regions is 1 + 2 + 2 + 1 + 1 = 7, so T(5,2) = 7.
_ _ _
|_| _|_| _|_| _ _ _ _
|_| |_|_| |_|_| |_|_| |_|_|
|_|_ |_| |_| |_|_| |_|_
|_|_| |_| |_| |_| |_|_|
.
For k = 3 there are six free pentominoes of width 3 as shown below, and from left to right there are respectively 3, 2, 1, 2, 4, 2 regions between the pentominoes and their bounding boxes, hence the total number of regions is 3 + 2 + 1 + 2 + 4 + 2 = 14, so T(5,3) = 14.
_ _ _ _ _ _ _ _ _ _
_|_|_| |_|_|_| |_| |_|_ _|_|_ |_|_|
|_|_| |_| |_|_ _ |_|_|_ |_|_|_| |_|_
|_| |_| |_|_|_| |_|_| |_| |_|_|
.
Therefore the 5th row of the triangle is [0, 7, 14].
.
A379629
Sum of the lengths of the free polyominoes with n cells.
Original entry on oeis.org
1, 2, 5, 15, 41, 139, 474, 1773, 6686, 26043, 101814, 402297, 1593124, 6332329, 25200575, 100462874, 400908688, 1601541747
Offset: 1
For n = 4 the free polyominoes with four cells are also called free tetrominoes.
The five free tetrominoes are as shown below:
_
_ _ _ |_|
_ _ |_| |_|_ |_|_ |_|
|_|_| |_|_ |_|_| |_|_| |_|
|_|_| |_|_| |_| |_| |_|
.
There are no free tetrominoes of length 1, there is only one free tetromino of length 2, there are three free tetrominoes of length 3 and there is only one free tetromino of length 4, hence the sum of the lengths is 0 + 2 + 3 + 3 + 3 + 4 = 15, so a(4) = 15.
A380285
Total number of regions between the free polyominoes with n cells and their bounding boxes.
Original entry on oeis.org
0, 0, 1, 5, 21, 71, 255, 961, 3630, 13973, 53938, 209641, 815784, 3183642, 12439291, 48686549, 190787588, 748645732
Offset: 1
Illustration for n = 4:
The free polyominoes with four cells are also called free tetrominoes.
The five free tetrominoes are as shown below:
_
|_| _ _ _
|_| |_| |_|_ |_|_ _ _
|_| |_|_ |_|_| |_|_| |_|_|
|_| |_|_| |_| |_| |_|_|
.
The bounding boxes are respectively as shown below:
_
| | _ _ _ _ _ _
| | | | | | | | _ _
| | | | | | | | | |
|_| |_ _| |_ _| |_ _| |_ _|
.
From left to right the number of regions between the free tetrominoes and their bounding boxes are respectively 0, 1, 2, 2, 0. Hence the total number of regions is 0 + 1 + 2 + 2 + 0 = 5, so a(4) = 5.
.
Cf.
A379628 (total area of the regions).
Cf.
A000105,
A057766,
A379623,
A379624,
A379625,
A379626,
A379627,
A379629,
A379637,
A379638,
A380282.
Showing 1-5 of 5 results.
Comments