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).

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

1,4

Comments

The width here is the shorter of the two dimensions.

Links

Table of n, a(n) for n=1..56.

Index entries for sequences related to polyominoes.

Formula

T(n,k) = k*A379623(n,k).

Example

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].
.

Crossrefs

Row lengths give A110654.

Row sums give A379626.

Cf. A000105, A379623, A379625, A379627, A379638.

Sequence in context: A102208 A371650 A369779 * A125911 A009385 A008308

Adjacent sequences: A379634 A379635 A379636 * A379638 A379639 A379640

Keyword

nonn,tabf,new

Author

Omar E. Pol, Jan 16 2025

Status

approved