A255890 -id:A255890 - cp's OEIS frontend

A367758 Least number of inequivalent cells in a polyomino with n cells.

1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 2, 4, 4, 5, 3, 4, 5, 6, 3, 5, 6, 7, 4, 5, 7, 8, 4, 6, 8, 9, 5, 6, 9, 10, 5, 7, 10, 11, 6, 7, 11, 12, 6, 8, 12, 13, 7, 8, 13, 14, 7, 9, 14, 15, 8, 9, 15, 16, 8, 10, 16, 17, 9, 10, 17, 18, 9, 11, 18, 19, 10, 11, 19, 20, 10, 12, 20, 21, 11, 12, 21, 22, 11, 13, 22, 23

Offset: 1

Pontus von Brömssen, Nov 29 2023

nonn

Two cells in a polyomino are equivalent if there is a symmetry of the polyomino that takes one of the cells to the other.

Equivalently, a(n) is the least number of pointed polyominoes (A126202) corresponding to a given polyomino with n cells.

			The X pentomino has 2 inequivalent cells and no pentomino have all cells equivalent, so a(5) = 2.

Pontus von Brömssen, Illustration of optimal polyominoes for n = 1..13.
John Mason and Pontus von Brömssen, Proof of formula.
Index entries for sequences related to polyominoes.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,0,1,-1,1,-1).

Cf. A126202, A255890, A367759, A369363, A369366, A376798.

a(n) > n/8.
From John Mason and Pontus von Brömssen, Oct 08 2024: (Start)
For n != 1,5, n = 8*k + c, for integers k and c, k >= 0, 0 <= c <= 7:
  if c = 0 or 1 then a(n) =   k + c + 1;
  if c = 2 or 6 then a(n) = 2*k + (c+2)/4;
  if c = 3 or 7 then a(n) = 2*k + (c+5)/4;
  if c = 4      then a(n) =   k + 1;
  if c = 5      then a(n) =   k + 3.
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-8) - a(n-9) + a(n-10) - a(n-11) for n >= 17.
a(n) = 2*a(n-8) - a(n-16) for n >= 22. (End)

a(14)-a(18) from John Mason, Sep 19 2024

More terms from John Mason, Oct 08 2024

A367443 a(n) is the number of free polyominoes that can be obtained from the polyomino with binary code A246521(n+1) by adding one cell.

Original entry on oeis.org

1, 2, 4, 3, 9, 1, 5, 4, 3, 8, 6, 5, 11, 10, 10, 6, 6, 9, 5, 2, 4, 5, 11, 13, 11, 3, 12, 9, 11, 10, 11, 5, 11, 5, 11, 12, 11, 12, 5, 6, 10, 5, 13, 12, 12, 7, 6, 6, 7, 11, 11, 6, 11, 6, 5, 4, 12, 11, 11, 13, 12, 11, 12, 14, 13, 12, 6, 7, 11, 3, 11, 11, 10, 11

Offset: 1

Pontus von Brömssen, Nov 18 2023

Can be read as an irregular triangle, whose m-th row contains A000105(m) terms, m >= 1.

			As an irregular triangle:
  1;
  2;
  4, 3;
  9, 1, 5,  4,  3;
  8, 6, 5, 11, 10, 10, 6, 6, 9, 5, 2, 4;
  ...
For n = 5, the L tetromino, whose binary code is A246521(5+1) = 15, can be extended to 9 different free pentominoes, so a(5) = 9. (All possible ways to add one cell lead to different pentominoes.)
For n = 6, the square tetromino, whose binary code is A246521(6+1) = 23, can only be extended to the P pentomino by adding one cell, so a(6) = 1.

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).
Index entries for sequences related to polyominoes.

Cf. A000105, A246521, A255890 (row minima), A367126, A367439, A367441.

A373635 Number of free n-celled polyominoes to which two inequivalent cells can be adjoined such that the two resulting free (n+1)-celled polyominoes are identical.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 0, 4, 6, 2, 0, 17, 29

Offset: 1

Pontus von Brömssen, Jun 13 2024

Two cells that can be adjoined to an n-celled polyomino are equivalent if there is an isomorphism between the two resulting (n+1)-celled polyominoes that maps the adjoined cell of the first to the adjoined cell of the second.

			The smallest example is the P pentomino shown below. The two free hexominoes obtained by adjoining one of the two cells marked "*" are identical, but there is no isomorphism between them that also maps the marked cell of the first to the marked cell of the second.
    _ _
  *|   |
   |  _|*
   |_|

Pontus von Brömssen, Illustration up to n = 11.
Index entries for sequences related to polyominoes.

Cf. A255890, A367758, A373636, A373637.

A255894 Polyiamond Family Planners: a(n) is the least number of children of a polyiamond of size n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 2, 2, 2, 5, 1, 3

Offset: 0

Gordon Hamilton, Mar 09 2015

			a(7) = 3 because the 7-triangle polyiamond
  ____________
  \          /\
   \        /  \
    \      /    \  a
     \    /      \
      \  /        \      b
       \/__________\___________
       /\          /\          /
      /  \        /  \        /
     /    \      /    \      /
    /      \    /      \    /  c
   /        \  /        \  /
  /__________\/__________\/
  \          /
   \        /
    \      /
     \    /
      \  /
       \/
has three 8-triangle children created by adding a triangle to position a, b or c.

Cf. A000577, A255890.

A359667 a(n) is the number of minimally prolific free polyominoes, i.e., that can generate the least possible number of children by adding a square.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 1, 5, 1, 1, 2, 2

Offset: 1

John Mason, Jan 10 2023

			a(8) = 5 because each of the following octominoes will generate the minimum number of children given by A255890(8) = 3:
  OO OOO  OO   O    O
  OO O O OOOO OOO   OOO
  OO OOO  OO   OOO OOO
  OO            O    O

John Mason, Illustrations

Cf. A000105, A255890.

cp's OEIS Frontend

A367758 Least number of inequivalent cells in a polyomino with n cells.

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A367443 a(n) is the number of free polyominoes that can be obtained from the polyomino with binary code A246521(n+1) by adding one cell.

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A373635 Number of free n-celled polyominoes to which two inequivalent cells can be adjoined such that the two resulting free (n+1)-celled polyominoes are identical.

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A255894 Polyiamond Family Planners: a(n) is the least number of children of a polyiamond of size n.

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A359667 a(n) is the number of minimally prolific free polyominoes, i.e., that can generate the least possible number of children by adding a square.

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Author

Keywords

Examples

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