Rodolfo Kurchan - cp's OEIS frontend

A380025 Area of smallest rectangle of grid cells such that it is possible to mark a connected subset of the cells so that the lengths of runs of marked cells have lengths from 2 to n, each length appearing exactly once.

2, 6, 12, 15, 24, 35, 40, 54, 65, 77, 96, 112, 126, 150, 160

Offset: 2

Rodolfo Kurchan, Jan 09 2025

Runs may be horizontal or vertical and only lengths > 1 are considered.
In the puzzle, the best solutions are the ones with the most free space (fewest marked cells). In this sequence, only the area of the smallest rectangle is considered.
Terms a(6)-a(10) from George Sicherman.
a(11) from Giorgio Vecchi.

			a(11) = 77 from the following 7 x 11 rectangle:
2 to 11 = 77 (7 x 11) (30 free space).
+---+---+---+---+---+---+---+---+---+---+---+
| X | X | X | X | X | X | X | X | X | X | X |
+---+---+---+---+---+---+---+---+---+---+---+
|   |   |   |   |   | X | X | X | X |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
| X | X | X | X | X | X | X | X | X | X |   |
+---+---+---+---+---+---+---+---+---+---+---+
|   |   |   |   |   | X | X |   | X |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
| X | X | X | X | X | X | X | X | X |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
|   |   |   |   |   | X |   |   | X |   |   |
+---+---+---+---+---+---+---+---+---+---+---+
| X | X | X | X | X | X | X | X |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+

Rodolfo Kurchan, Crossword Polyominoes, Puzzle Fun, Problems (2025).

Cf. A351516.

a(12)-a(16) from Rodolfo Kurchan, Feb 01 2025

A379589 Maximum number of connections for a 4 X n rectangle.

Original entry on oeis.org

1, 31, 800, 6466, 60778, 441492, 3216584, 18693320

Offset: 2

Rodolfo Kurchan, Dec 26 2024

In a 4 X n board (with n > 1) with numbers 1, 2 3 and 4, at least 2 of each, find the arrangement with more solutions connecting a pair of numbers 1, a pair of number 2, a pair of number 3 and a pair of number 4, covering the entire board and without passing through the same square twice.

Terms a(5)-a(9) from Giorgio Vecchi.

			For n = 2 with the board
+---+---+
| 1 | 1 |
+---+---+
| 2 | 2 |
+---+---+
| 3 | 3 |
+---+---+
| 4 | 4 |
+---+---+
There is only 1 solution being the squares with these letters:
+---+---+
| A | B |
+---+---+
| C | D |
+---+---+
| E | F |
+---+---+
| G | H |
+---+---+
Solution:
1) AB - CD - EF - GH
There is one solution so a(2) = 1.
.
For n = 3 with the board
+---+---+---+
| 1 | 2 | 2 |
+---+---+---+
| 1 | 2 | 2 |
+---+---+---+
| 3 | 4 | 4 |
+---+---+---+
| 3 | 4 | 4 |
+---+---+---+
the maximum number of solutions is 31 being the squares with these letters:
+---+---+---+
| A | B | C |
+---+---+---+
| D | E | F |
+---+---+---+
| G | H | I |
+---+---+---+
| J | K | L |
+---+---+---+
Solutions:
  1)	AD - GJ - BC - HEFILK
  2)	AD - GJ - BC - IFEHKL
  3)	AD - GJ - BC - KHEFIL
  4)	AD - GJ - KL - EHIFCB
  5)	AD - GJ - KL - FIHEBC
  6)	AD - GJ - KL - BEHIFC
  7)	AD - GJ - BCFE - HILK
  8)	AD - GJ - BCFE - ILKH
  9)	AD - GJ - BCFE - LKHI
  10)	AD - GJ - BCFE - KHIL
  11)	AD - GJ - CFEB - HILK
  12)	AD - GJ - CFEB - ILKH
  13)	AD - GJ - CFEB - LKHI
  14)	AD - GJ - CFEB - KHIL
  15)	AD - GJ - FEBC - HILK
  16)	AD - GJ - FEBC - ILKH
  17)	AD - GJ - FEBC - LKHI
  18)	AD - GJ - FEBC - KHIL
  19)	AD - GJ - EBCF - HILK
  20)	AD - GJ - EBCF - ILKH
  21)	AD - GJ - EBCF - LKHI
  22)	AD - GJ - EBCF - KHIL
  23)	ABED - GJ - CF - HILK
  24)	ABED - GJ - CF - ILKH
  25)	ABED - GJ - CF - LKHI
  26)	ABED - GJ - CF - KHIL
  27)	GHKJ - AD - IL - BCFE
  28)	GHKJ - AD - IL - CFEB
  29)	GHKJ - AD - IL - FEBC
  30)	GHKJ - AD - IL - EBCF
  31)	ABED - GHKJ - CF - IL
There are 31 solutions so a(3) = 31.

Rodolfo Kurchan and Claudio Meller, Number Connections, Puzzle Fun, Problems (2024).

Cf. A379241, A379393.

A379393 Maximum number of connections for a 3 X n rectangle.

Original entry on oeis.org

1, 6, 72, 277, 1910, 8657, 27442, 97132, 295752, 967914, 2922814

Offset: 2

Rodolfo Kurchan, Dec 22 2024

In a 3 X n board (with n > 1) with numbers 1, 2 and 3, at least 2 of each, find the arrangement with more solutions connecting a pair of numbers 1 and a pair of number 2 and a pair of number 3, covering the entire board and without passing through the same square twice.

Terms a(5) and a(7)-a(12) from Giorgio Vecchi.

			For n = 2 with the board
  +---+---+
  | 1 | 1 |
  +---+---+
  | 2 | 2 |
  +---+---+
  | 3 | 3 |
  +---+---+
There is only 1 solution being the squares with these letters:
  +---+---+
  | A | B |
  +---+---+
  | C | D |
  +---+---+
  | E | F |
  +---+---+
Solution:
1) AB - CD - EF
There is one solution so a(2) = 1.
.
For n = 3 with the board
  +---+---+---+
  | 1 | 3 | 3 |
  +---+---+---+
  | 1 | 2 | 2 |
  +---+---+---+
  | 1 | 2 | 2 |
  +---+---+---+
the maximum number of solutions is 6 being the squares with this letters:
  +---+---+---+
  | A | B | C |
  +---+---+---+
  | D | E | F |
  +---+---+---+
  | G | H | I |
  +---+---+---+
Solutions:
  1) ADG - BC - HEFI
  2) ADG - BC - FEHI
  3) ADG - BC - EFIH
  4) ADG - BC - EHIF
  5) ADG - BEFC - HI
  6) ADEHG - BC - FI
There are six solutions so a(3) = 6.

Rodolfo Kurchan and Claudio Meller, Number Connections, Puzzle Fun, Problems (2024).

Cf. A379241.

A379241 Maximum number of connections for a 2 X n rectangle.

Original entry on oeis.org

1, 5, 22, 54, 134, 269, 534, 934, 1618, 2573, 4062, 6030, 8902, 12549, 17614, 23854, 32194, 42229, 55238

Offset: 2

Rodolfo Kurchan, Dec 18 2024

In a 2 X n board (with n > 1) with numbers 1 and 2, at least 2 of each, find the arrangement with more solutions connecting a pair of numbers 1 and a pair of number 2, covering the entire board and without passing through the same square twice.
Terms a(7)-a(20) from Giorgio Vecchi.
In a vertex-colored graph, partition the vertices into paths of positive lengths. It is required that the two terminal vertices of each path have the same color, and that there is exactly one such path for each color of the terminal vertices. a(n) is the maximum number of such partitions for all possible 2-colorings (at least two vertices of each color) of the 2 X n grid graph. - Pontus von Brömssen, Dec 19 2024

			For n = 3 the with the board
  +---+---+---+
  | 1 | 1 | 2 |
  +---+---+---+
  | 1 | 2 | 2 |
  +---+---+---+
the maximum number of solutions is 5 being the squares with this letters:
  +---+---+---+
  | A | B | C |
  +---+---+---+
  | D | E | F |
  +---+---+---+
Solutions:
  1) ABED - CF
  2) ADEB - CF
  3) BAD - CFE
  4) AD - CBEF
  5) AD - EBCF
There are five solutions so a(3) = 5.

Rodolfo Kurchan and Claudio Meller, Number Connections, Puzzle Fun, Problems (2024).

A375507 a(1) = 1. For n > 1; a(n) is equal to a(n-1) plus the decimal value of the concatenation of the first n-1 digits of the sequence.

Original entry on oeis.org

1, 2, 14, 135, 1349, 13490, 134903, 1349038, 13490389, 134903902, 1349039036, 13490390385, 134903903876, 1349039038789, 13490390387923, 134903903879272, 1349039038792762, 13490390387927663, 134903903879276676, 1349039038792766810, 13490390387927668159

Offset: 1

Rodolfo Kurchan, Aug 18 2024

			For n = 4 we have that a(n-1) = a(3) = 14 and the decimal value of the concatenation of the first three digits of the sequence is 121, so a(4) = 14 + 121 = 135.

from itertools import count
def A375507_list(nmax):
    a = [1]
    def digits():
        for i in count():
            for d in str(a[i]):
                yield int(d)
    diff = 0
    for n,d in enumerate(digits(),1):
        if n==nmax: return a
        diff = 10*diff+d
        a.append(a[-1]+diff) # Pontus von Brömssen, Aug 18 2024

a(n) ~ (c/9)*10^(n-1), where c = Sum_{n>=1} a(n)/10^(n*(n-1)/2) = 1.2141351349... . - Pontus von Brömssen, Aug 18 2024

a(17)-a(21) from Pontus von Brömssen, Aug 18 2024

A371476 a(n) is the number of free polyominoes of size n with at least one solution to the One Up puzzle (see comments).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 10, 12, 23, 35, 169, 255, 817, 1883, 4702, 10489

Offset: 1

Rodolfo Kurchan, Mar 24 2024

The objective of the One Up puzzle is to assign a positive integer to each cell of a given polyomino in such a way that the cells of any maximal 1 X k strip (horizontal or vertical) are numbered 1, ..., k (in some order). The maximality is applied to horizontal and vertical strips separately, implying that the number 1 must be assigned to a cell with no left or right neighbors even if it has neighbors above or below (and vice versa). (In an extended version of the puzzle, there may be walls between certain pairs of neighboring cells, and only those strips that do not extend over a wall are considered.) - Pontus von Brömssen, Mar 26 2024

			The a(6) = 3 solvable hexominoes have unique solutions:
  +---+           +---+---+       +---+---+
  | 1 |           | 1 | 2 |       | 2 | 1 |
  +---+---+       +---+---+---+   +---+---+---+
  | 2 | 1 |       | 2 | 3 | 1 |   | 3 | 2 | 1 |
  +---+---+---+   +---+---+---+   +---+---+---+
  | 3 | 2 | 1 |       | 1 |       | 1 |
  +---+---+---+       +---+       +---+
a(7) = 10 because there are 10 heptominoes that have at least one solution to the One Up puzzle.

Rodolfo Kurchan, One Up Puzzle.
Rodolfo Kurchan, Puzzle Fun (see One Up).

Cf. A000105, A371828 (a generalization to hypergraphs).

a(7)-a(14) from Pontus von Brömssen, Mar 25 2024
a(15) from Pontus von Brömssen, Mar 26 2024
a(16) from Pontus von Brömssen, Apr 04 2024

A365906 Irregular triangle T(n,k) read by rows, n>=1, k>=1, in which row n lists in nonincreasing order the sum of the b values (described in A365835) of the cells of every free polyomino with n cells.

Original entry on oeis.org

1, 4, 9, 7, 16, 12, 12, 12, 10, 25, 19, 19, 17, 17, 17, 17, 15, 15, 15, 15, 13, 36, 28, 28, 28, 24, 24, 24, 24, 24, 24, 24, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 20, 20, 20, 20, 20, 20, 20, 18, 18, 18, 18, 18, 16, 49, 39, 39, 39, 33, 33, 33, 33, 33, 33

Offset: 1

Rodolfo Kurchan and Omar E. Pol, Sep 22 2023

Observation: at least for 1 <= n <= 6 the parity of the terms in row n coincides with the parity of n. If n is odd then every polyomino has an odd number of odd b values, otherwise if n is even then every polyomino has an even number of odd b values.

The preceding observation is true for all n, because the b-values count each cell once (it is in the same row/column as itself) and pairs of distinct cells in the same row or column (with no gaps in between) twice (once in each direction). - Pontus von Brömssen, Oct 15 2023

			Triangle begins:
   1;
   4;
   9,  7;
  16, 12, 12, 12, 10;
  25, 19, 19, 17, 17, 17, 17, 15, 15, 15, 15, 13;
  ...
For n = 5 the twelve pentominoes and the b values of their cells are as shown below:
.
      I     L         Y      P        T       V           X
.     _     _         _     _ _     _ _ _     _           _
     |_|   |_|      _|_|   |_|_|   |_|_|_|   |_|        _|_|_
     |_|   |_|     |_|_|   |_|_|     |_|     |_|_ _    |_|_|_|
     |_|   |_|_      |_|   |_|       |_|     |_|_|_|     |_|
     |_|   |_|_|     |_|
     |_|
      5     4         4     4 3     3 5 3     3           3
      5     4       2 5     4 3       3       3         3 5 3
      5     4         4     3         3       5 3 3       3
      5     5 2       4
      5
.
      F        N        U         Z         W
.     _ _       _     _   _     _ _       _
    _|_|_|    _|_|   |_|_|_|   |_|_|     |_|_
   |_|_|     |_|_|   |_|_|_|     |_|_    |_|_|_
     |_|     |_|                 |_|_|     |_|_|
             |_|
      4 2       2     2   2     2 4       2
    2 4       4 3     4 3 4       3       3 3
      3       3                   4 2       3 2
              3
.
T(5,k) is the sum of the b values of all cells of the k-th pentomino from the diagram.
For further information see also A365835.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (first 10 rows)
Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes
George Sicherman, A colored version of the free pentominoes

Row lengths give A000105, n >= 1.
Right border gives A016777.
Row sums give A365835.
Cf. A000035, A000290, A002378, A057766, A117950, A279019, A365860.

For n >= 1; T(n,1) = n^2.
For n >= 3; T(n,2) = (n - 1)^2 + 3 = A117950(n-1).
For n >= 4; T(n,3) = (n - 1)^2 + 3 = A117950(n-1).

Terms a(61) and beyond from Pontus von Brömssen, Oct 15 2023

A365860 For every cell of a polyomino let c be the number of cells that are in the same row or in the same column (including itself). a(n) is the sum of the c values of all cells of all free polyominoes with n cells.

Original entry on oeis.org

1, 4, 16, 62, 206, 790, 3042, 12648, 52181, 220372, 927333, 3917738, 16491489, 69356256, 290882884, 1217780926

Offset: 1

Rodolfo Kurchan and Omar E. Pol, Sep 20 2023

For a polyomino with n cells the maximum sum possible of the c values equals n^2 = A000290(n) and the minumum sum possible of the c values equals 3*(n - 2) + 4 = A016777(n-1). Hence the difference between the maximum possible and the minimum possible sum of the c values is A000290(n) - A016777(n-1) = A279019(n+3), n >= 1. Also it's equal to A002378(n-1) if n >= 2. See examples.
Note that the concept "c value" for a cell or vertex can also be applied in other polyforms and in other types of graphs, for example: cellular automata, partitions, etc.
For another version and further information see A365835, which first differs at a(5).

			For n = 1 the monomino has only one cell, so a(1) = 1.
For n = 2 the domino has two cells. Each cell sees the other cell. The sum of the c values is 2 + 2 = 4, so a(2) = 4.
For n = 3 the sum of the c values of the I-tromino is 3 + 3 + 3 = 9 and the sum of the c values of the L-tromino is 3 + 2 + 2 = 7. The total sum of the c values is 9 + 7 = 16, so a(3) = 16.
For n = 4 the c values of the five (I, L, O, T, S) tetrominoes are 16, 12, 12, 12, 10 so the total sum of the c values is a(4) = 62.
Three examples from the twelve pentominoes:
The I-pentomino with its c values looks like this:
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
The sum of the c values is 5 + 5 + 5 + 5 + 5 = 5^2 = 25, the maximum possible.
.
The U-pentomino with its c values looks like this:
  +---+   +---+
  | 3 |   | 3 |
  +---+---+---+
  | 4 | 3 | 4 |
  +---+---+---+
The sum of the c values is 4 + 4 + 3 + 3 + 3 = 17.
.
The W-pentomino with its c values looks like this:
  +---+
  | 2 |
  +---+---+
  | 3 | 3 |
  +---+---+---+
      | 3 | 2 |
      +---+---+
The sum of the c values is 3 + 3 + 3 + 2 + 2 = 3*(5-2) + 4 = 13, the minimum possible.
.

Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes

Cf. A000105, A000290, A002378, A016777, A057766, A279019, A365835.

a(n) == A057766(n) (mod 2). - Pontus von Brömssen, Sep 21 2023

a(7)-a(9) from George Sicherman, Sep 20 2023
a(10)-a(13) from Pontus von Brömssen, Sep 21 2023
a(14)-a(16) from Pontus von Brömssen, Apr 03 2024

A365835 For each cell of a polyomino let b be the number of cells that are in the same row or in the same column (including itself). Cells beyond gaps do not count. a(n) is the sum of the b values of all cells of all free polyominoes with n cells.

Original entry on oeis.org

1, 4, 16, 62, 204, 776, 2936, 12030, 48783, 202734, 839239, 3489810, 14462593, 59906626, 247553908, 1021545890

Offset: 1

Rodolfo Kurchan and Omar E. Pol, Sep 19 2023

The sum of the b values of a polyomino seems to give an idea of its "alignment". It seems the highest values correspond to the most aligned polyominoes and the lowest values correspond to the least aligned polyominoes. For example, in an I-polyomino with n cells the sum of the b values equals n^2 = A000290(n), which is the maximum possible. Other polyominoes with k cells have a lower value.
A question from Jon E. Schoenfield: Is it true that the minimum sum of the b values (for a given value of n, the number of cells) is always obtained by only one polyomino, and that that is the one that can be built using a tight zigzag pattern (turning alternately to the left or right at each step) -- i.e., the monomino, the domino, the "L"-tromino, the "S"-tetromino, the "W"-pentomino, etc.?
The answer is: Yes. And the sum of the b values is equal to 3*(n - 2) + 4 = A016777(n-1), the minimum possible.
Hence the difference between the maximum possible and the minimum possible sum of the b values is A000290(n) - A016777(n-1) = A279019(n+3), n >= 1. Also it's equal to A002378(n-1) if n >= 2. See examples.
Resembles the art gallery problem.
Note that the concept "b value" for a cell or vertex can also be applied in other polyforms and in other types of graphs, for example: cellular automata, partitions, etc.
For another version see A365860, which first differs at a(5).

			For n = 1 the monomino has only one cell, so a(1) = 1.
For n = 2 the domino has two cells. Each cell sees the other cell. The sum of the b values is 2 + 2 = 4, so a(2) = 4.
For n = 3 the sum of the b values of the I-tromino is 3 + 3 + 3 = 9 and the sum of the b values of the L-tromino is 3 + 2 + 2 = 7. The total sum is 9 + 7 = 16, so a(3) = 16.
For n = 4 the b values of the five tetrominoes (I, L, O, T, S) are 16, 12, 12, 12, 10, so the total sum of the b values is a(4) = 62.
Three examples from the twelve pentominoes:
The I-pentomino with its b values looks like this:
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
      | 5 |
      +---+
The sum of the b values is 5 + 5 + 5 + 5 + 5 = 5^2 = A000290(5) = 25, the maximum possible.
.
The U-pentomino with its b values looks like this:
  +---+   +---+
  | 2 |   | 2 |
  +---+---+---+
  | 4 | 3 | 4 |
  +---+---+---+
The sum of the b values is 4 + 4 + 3 + 2 + 2 = 15.
.
The W-pentomino with its b values looks like this:
  +---+
  | 2 |
  +---+---+
  | 3 | 3 |
  +---+---+---+
      | 3 | 2 |
      +---+---+
The sum of the b values is 3 + 3 + 3 + 2 + 2 = 3*(5-2) + 4 = A016777(5-1) = 13, the minimum possible.
.

Rodolfo Kurchan, Puzzle Fun, Problems, Colored Polyominoes.
George Sicherman, A colored version of the free pentominoes.
Wikipedia, Polyomino.
Wikipedia, Art gallery problem.

Row sums of A365906.

Cf. A000105, A000290, A002378, A016777, A057766, A279019, A365860.

a(n) == A057766(n) (mod 2). - Pontus von Brömssen, Sep 21 2023

a(6)-a(9) from George Sicherman, Sep 20 2023
a(6)-a(9) corrected and a(10)-a(13) added by Pontus von Brömssen, Sep 21 2023
a(14)-a(16) from Pontus von Brömssen, Apr 03 2024

A362601 Domination number for pawns' graph P(n).

Original entry on oeis.org

1, 2, 5, 8, 12, 16, 23, 28, 33, 44, 49, 56, 70, 78, 85, 104, 111, 120, 141, 152, 161, 188, 197, 208, 237, 250, 261, 296, 307, 320, 357, 372, 385, 428, 441, 456, 501, 518, 533, 584

Offset: 1

Rodolfo Kurchan, Jun 18 2023

Minimum number of white pawns needed to occupy or attack all squares of an n X n chessboard.
Solutions for boards of sizes 1 to 8, 10, 14, 15 from Rodolfo Kurchan.
Solutions for boards of sizes 9, 11, 12, 14 to 18, 20 to 24 from Michael Steinau.
Solution for boards of size 13 and 25 to 40 from M. Achterberg.

			a(8) = 28 white pawns occupying or attacking all squares of a standard chessboard:
  . . . . . . . .
  . P P P P P P .
  . P . . . . P .
  . P . P P . P .
  . P . . . . P .
  . P . P P . P .
  . P . . . . P .
  P P P P P P P P

M. Achterberg, Illustration for a(13) = 70.
Rodolfo Kurchan, Board domination problem, Chess Puzzle Fun 6, April 2023.

Cf. A075458.

cp's OEIS Frontend

User: Rodolfo Kurchan

A380025 Area of smallest rectangle of grid cells such that it is possible to mark a connected subset of the cells so that the lengths of runs of marked cells have lengths from 2 to n, each length appearing exactly once.

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A379589 Maximum number of connections for a 4 X n rectangle.

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A379393 Maximum number of connections for a 3 X n rectangle.

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A379241 Maximum number of connections for a 2 X n rectangle.

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A375507 a(1) = 1. For n > 1; a(n) is equal to a(n-1) plus the decimal value of the concatenation of the first n-1 digits of the sequence.

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A371476 a(n) is the number of free polyominoes of size n with at least one solution to the One Up puzzle (see comments).

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A365906 Irregular triangle T(n,k) read by rows, n>=1, k>=1, in which row n lists in nonincreasing order the sum of the b values (described in A365835) of the cells of every free polyomino with n cells.

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A365860 For every cell of a polyomino let c be the number of cells that are in the same row or in the same column (including itself). a(n) is the sum of the c values of all cells of all free polyominoes with n cells.

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A365835 For each cell of a polyomino let b be the number of cells that are in the same row or in the same column (including itself). Cells beyond gaps do not count. a(n) is the sum of the b values of all cells of all free polyominoes with n cells.

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