A352720 -id:A352720 - cp's OEIS frontend

A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).

1, 1, 1, 1, 1, 4, 1, 5, 6, 1, 12, 22, 1, 18, 71, 18, 1, 37, 193, 138, 1, 60, 490, 661, 73, 1, 117, 1221, 2547, 769, 1, 200, 3011, 8417, 5189, 255, 1, 379, 7393, 26164, 25920, 3743, 1, 669, 18025, 78074, 108834, 32038, 950, 1, 1250, 43847, 229881, 408217, 201956, 16819

Offset: 1

Omar E. Pol, Jan 07 2025

The width here is the shorter of the two dimensions.

			Triangle begins:
  1;
  1;
  1,    1;
  1,    4;
  1,    5,     6;
  1,   12,    22;
  1,   18,    71,     18;
  1,   37,   193,    138;
  1,   60,   490,    661,     73;
  1,  117,  1221,   2547,    769;
  1,  200,  3011,   8417,   5189,    255;
  1,  379,  7393,  26164,  25920,   3743;
  1,  669, 18025,  78074, 108834,  32038,   950;
  1, 1250, 43847, 229881, 408217, 201956, 16819;
  ...
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, so T(5,2) = 5.
   _           _         _
  |_|        _|_|      _|_|      _ _       _ _
  |_|       |_|_|     |_|_|     |_|_|     |_|_|
  |_|_      |_|         |_|     |_|_|     |_|_
  |_|_|     |_|         |_|     |_|       |_|_|
.
For k = 3 there are six free pentominoes of width 3 as shown below, so T(5,3) = 6.
     _ _     _ _ _     _         _           _       _ _
   _|_|_|   |_|_|_|   |_|       |_|_       _|_|_    |_|_|
  |_|_|       |_|     |_|_ _    |_|_|_    |_|_|_|     |_|_
    |_|       |_|     |_|_|_|     |_|_|     |_|       |_|_|
.
Therefore the 5th row of the triangle is [1, 5, 6] and the row sum is A000105(5) = 12.
.

John Mason, Table of n, a(n) for n = 1..90 (first 18 rows)
Index entries for sequences related to polyominoes.

Row sums give A000105(n).
Row lengths give A110654(n).
For free polyominoes of length k see A379624.
Cf. A057051, A352720, A379627, A379628.

a(21)-a(56) from Pontus von Brömssen, Jan 11 2025

A335711 The number of free polyominoes of width 2 and height n.

Original entry on oeis.org

2, 6, 12, 30, 65, 158, 362, 875, 2064, 4984, 11914, 28764, 69155, 166956, 402372, 971413, 2343518, 5657754, 13654968, 32966010, 79577189, 192116330, 463786190, 1119678911, 2703086892, 6525829036, 15754607062, 38034986040, 91824246215, 221683340568, 535190123592

Offset: 2

R. J. Mathar, Jun 18 2020

nonn

The second column of A268371.

			a(2)=2, bounding box 2 X 2, counts the L-shaped 3-omino and the full block 4-omino.
a(3)=6, bounding box 2 X 3, counts three 4-ominoes, two 5-omioes, and the full 2 X 3 block 6-omino.
a(4)=12, bounding box 2 X 4, counts three 5-ominoes, six 6-ominoes, two 7-ominoes, and the full 2 X 4 block 8-omino.

Cf. A268371, A107769 (asymmetric), A005409 (C_2 symmetry and higher), A352720 (width 2 and size n).

Conjecture: a(n) = A107769(n-1) + A005409(floor((n+3)/2)).
Conjectures from Colin Barker, Jun 24 2020: (Start)
G.f.: x^2*(2 - 8*x^2 + 2*x^3 - x^4 + x^5 + x^6) / ((1 - x)*(1 - 2*x - x^2)*(1 - 2*x^2 - x^4)).
a(n) = 3*a(n-1) + a(n-2) - 7*a(n-3) + 3*a(n-4) - a(n-5) + a(n-6) + a(n-7) for n>8.
(End)
a(n) = (2*r(n) + 2*m(n) + A078057(n) + 1) / 4, where r(n) = A078057(floor((n-1)/2) - 1)/2, and m(n) = A078057(floor((n+1)/2) - 3)/2. - John Mason, Feb 28 2022

a(12)-a(20) from Jean-Luc Manguin, Jun 23 2020
a(21)-a(28) from John Mason, Feb 27 2022
a(29)-a(32) from John Mason, Feb 28 2022

A353067 a(n) is the number of free polyominoes of width 3 and size n.

Original entry on oeis.org

1, 3, 8, 23, 71, 193, 490, 1221, 3011, 7393, 18025, 43847, 106206, 256851, 619642, 1493272, 3593527, 8641624, 20764212, 49869733, 119716469, 287306693, 689316717, 1653539600, 3965895220, 9510880953, 22806565004, 54685294862, 131116579494, 314360235644

Offset: 3

John Mason, Apr 21 2022

nonn

The sequence can be generated using a series of recursive formulas in a fashion similar to A352720. This latter requires around 500 formula lines, whereas a(n) requires about 62000 lines.

			a(3) = 1 because of the straight tromino.
a(4) = 3 because of the following tetrominoes:
  O     O    OO
  OOO  OOO  OO

John Mason, Table of n, a(n) for n = 3..400
John Mason, Java program containing formulas

Cf. A000105, A352720 (width 2 and size n), A354850 (width 4 and size n).

A354850 a(n) is the number of free polyominoes of width 4 and size n.

Original entry on oeis.org

1, 3, 21, 59, 198, 703, 2568, 8421, 26165, 78074, 229881, 668082, 1928220, 5523946, 15745682, 44666804, 126251748, 355692380, 999498933, 2802212026, 7841587533, 21907928927, 61123152811, 170333279738, 474197771123, 1319004492132, 3666181193067, 10183729521212

Offset: 4

John Mason, Jun 08 2022

nonn

The sequence can be generated using a series of recursive formulas in a fashion similar to A352720 and A353067. These require respectively around 500 and 62000 formula lines, whereas a(n) requires about 10 million lines.

John Mason, Table of n, a(n) for n = 4..100
John Mason, Java programs for calculating sequence, valid also for widths 2 and 3

Cf. A000105, A352720, A353067.

cp's OEIS Frontend

A379623 Irregular triangle read by rows: T(n,k) is the number of free polyominoes with n cells and width k, n >= 1, 1 <= k <= ceiling(n/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A335711 The number of free polyominoes of width 2 and height n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A353067 a(n) is the number of free polyominoes of width 3 and size n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A354850 a(n) is the number of free polyominoes of width 4 and size n.

Views

Author

Keywords

Comments

Links

Crossrefs