A071764 -id:A071764 - cp's OEIS frontend

A381703 Irregular triangle read by rows in which every row of length A071764(n) lists A(n,w,h) = the number of free polyominoes of size n, width w and height h (for w <= h, and all possible w,h pairs).

1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 6, 1, 1, 6, 5, 7, 15, 1, 2, 11, 5, 7, 39, 25, 18, 1, 1, 10, 19, 7, 3, 59, 96, 35, 77, 61, 1, 3, 22, 28, 7, 1, 42, 210, 188, 49, 181, 383, 97, 73, 1, 1, 15, 52, 40, 9, 21, 255, 550, 332, 63, 266, 1304, 822, 155, 529, 240, 1, 3, 45, 90, 53, 9, 4, 212, 954, 1231, 529, 81, 251, 2847, 3548, 1551, 220, 2413, 2366, 410, 255

Offset: 1

John Mason, Mar 04 2025

			Triangle begins:
   n
   1:  1
   2:  1
   3:  1  1
   4:  1  1   3
   5:  1  2   3   6
   6:  1  1   6   5   7  15
   7:  1  2  11   5   7  39  25   18
   8:  1  1  10  19   7   3  59   96   35   77   61
   9:  1  3  22  28   7   1  42  210  188   49  181  383    97   73
  10:  1  1  15  52  40   9  21  255  550  332   63  266  1304  822  155  529  240
  ...
Any row contains an irregular array that shows the number of polyominoes having width w and height h. E.g., row 6 contains the array:
  h/w 1  2  3
  1
  2
  3      1  7
  4      6 15
  5      5
  6   1
.
There are 5 polyominoes of size 6 with width 2 and height 5, so A(6,2,5)=5:
.
  OO O  O  O  O
  O  OO O  O  O
  O  O  OO O  OO
  O  O  O  OO  O
  O  O  O   O  O

John Mason, Table of n, a(n) for n = 1..386

Row sums give A000105.
Row lengths give A071764.
Cf. A379623, A379624, A379625, A317186.

More terms from John Mason, Mar 07 2025

A283106 Number of distinct envelope areas of the polyominoes of order n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 9, 11, 12, 16, 18, 22, 26, 30, 33, 37, 41, 47, 52, 58, 62, 69, 74, 80, 86, 92, 97, 106, 112, 120, 127, 135, 143, 152, 159, 168, 176, 186, 193, 205, 213, 225, 235, 246, 256, 269, 281, 291, 302

Offset: 0

Francois Alcover, Feb 28 2017

nonn

The envelope of a polyomino is the minimal rectangle containing it.

			a(3) = 2;
The two possible envelopes are
|*|
|*|
|*|    area = 3;
and
|*| |
|*|*|  area = 4.

Cf. A071764 (distinct envelope ratios).

cp's OEIS Frontend

A381703 Irregular triangle read by rows in which every row of length A071764(n) lists A(n,w,h) = the number of free polyominoes of size n, width w and height h (for w <= h, and all possible w,h pairs).

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