A291809 -id:A291809 - cp's OEIS frontend

A291806 The number of polyomino tilings of n X n square.

1, 5, 216, 212987

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

Using the same terminology used for polyominoes: the corresponding sequence for "fixed" tilings is 1,12,1434,1691690, and for one-sided tilings is 1,5,222,213315.

John Mason, Tiling examples

Cf. A268416 (polyominoes that will fit in n-sided square), A291807 (symmetric tilings), A291808 (tilings with distinct polyominoes), A291809 (tilings with differently sized polyominoes).

A291808 Number of tilings of an n X n square using distinct polyominoes.

Original entry on oeis.org

1, 2, 44, 10066

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that are formed of distinct polyominoes. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

John Mason, Example tilings

Cf. A268416 (polyominoes that will fit in n-sided square), A291806 (polyomino tilings of square), A291807 (symmetric tilings), A291809 (tilings with differently sized polyominoes).

A291807 The number of symmetric polyomino tilings of n X n square.

Original entry on oeis.org

1, 5, 67, 3000

Offset: 1

John Mason, Sep 01 2017

The sequence gives the number of distinct tilings by polyominoes of a square with side n, considering tilings that have at least one symmetry. As for "free" polyominoes, tilings that are reflections or rotations of each other are not considered distinct.

John Mason, Example symmetric tilings

Cf. A268416 (polyominoes that will fit in n-sided square), A291806 (polyomino tilings of square), A291808 (tilings with distinct polyominoes), A291809 (tilings with differently sized polyominoes).

A347581 The Barnyard sequence: a(n) is the minimum number of unit length line segments required to enclose areas of 1 through n on a square grid.

Original entry on oeis.org

4, 9, 14, 20, 26, 33, 40, 47, 55, 63

Offset: 1

Scott R. Shannon, Oct 05 2021

The areas of size 1 through n can be created in any order and position, the only requirement being the final number of line segments used to enclose all areas is minimized. It is likely the perimeter of each area of size k, 1 <= k <= n, is the minimum possible for an area of size k, although this is unknown.

See A348149 for the total segments when the number of segments at each step is minimized.

			Example areas using the minimum number of line segments from n = 1 through n = 10 are:
.
   __
  |__|  a(1) = 4
   __ __ __
  |__|__ __|  a(2) = 9
   __ __ __
  |__|__ __|  a(3) = 14
  |__ __ __|
   __ __ __
  |__|__ __|
  |__ __ __|  a(4) = 20
  |     |
  |__ __|
   __ __ __
  |__|__ __|__
  |__ __ __|  |  a(5) = 26
  |     |     |
  |__ __|__ __|
   __ __ __
  |__|__ __|__ __ __
  |__ __ __|  |     |  a(6) = 33
  |     |     |     |
  |__ __|__ __|__ __|
         __ __ __ __
   __ __|__         |
  |__|__ __|__ __ __|
  |__ __ __|  |     |  a(7) = 40
  |     |     |     |
  |__ __|__ __|__ __|
   __ __ __ __ __ __
  |           |     |
  |__ __ __ __|     |
  |        |__ __ __|   a(8) = 47
  |__ __ __|__      |
  |     |  |  |__ __|
  |__ __|__|__ __|__|
   __ __ __ __ __ __ __
  |        |           |
  |        |__ __ __ __|
  |__ __ __|__         |
     |__|__ __|__ __ __|  a(9) = 55
     |__ __ __|  |     |
     |     |     |     |
     |__ __|__ __|__ __|
   __ __ __ __ __ __ __ __
  |         __|__   |     |
  |__ __ __|     |__|__   |
  |        |     |     |__|
  |        |     |     |  |   a(10) = 63
  |__ __ __|__ __|__ __|__|
  |              |     |__|
  |__ __ __ __ __|__ __|
.

Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2015.

Cf. A348149, A001168, A291808, A291809, A328020, A291806, A172477, A006983.

A348149 Variation of the Barnyard sequence A347581: a(n) is the minimum number of unit-length line segments required to enclose areas of 1 through n on a square grid when the number of segments is minimized as each area of incrementing size, starting at 1, is added.

Original entry on oeis.org

4, 9, 14, 20, 26, 33, 40, 48, 55, 64

Offset: 1

Scott R. Shannon, Oct 03 2021

In this variation of A347581 the areas must be added in the order of their sizes, from 1 through n, and as each area is added the minimum possible number of line segments must be used. This forces, for example, the first three areas of size 1, 2 and 3 to form a 2 X 3 block and thus they can never appear in any other arrangement in the final area. This is also true for n up to at least 9 due to the restriction of maximizing the usable edges for the next area. This leads to a(8) and a(10) containing one more line segment than the optimal solutions of A347581.

			Examples of n = 1 to n = 10 are given below. Note that for a(3) the configuration could also consist of the area of size 1 sitting above the area of size 2 with the area of size 3 forming an L-shaped block creating the minimal 2 X 3 block.
.
   __
  |__|  a(1) = 4
   __ __ __
  |__|__ __|  a(2) = 9
   __ __ __
  |__|__ __|  a(3) = 14
  |__ __ __|
   __ __ __
  |__|__ __|
  |__ __ __|  a(4) = 20
  |     |
  |__ __|
   __ __ __
  |__|__ __|__
  |__ __ __|  |  a(5) = 26
  |     |     |
  |__ __|__ __|
   __ __ __
  |__|__ __|__ __ __
  |__ __ __|  |     |  a(6) = 33
  |     |     |     |
  |__ __|__ __|__ __|
         __ __ __ __
   __ __|__         |
  |__|__ __|__ __ __|
  |__ __ __|  |     |  a(7) = 40
  |     |     |     |
  |__ __|__ __|__ __|
         __ __ __ __
        |           |
        |__ __ __ __|
   __ __|__         |
  |__|__ __|__ __ __|  a(8) = 48
  |__ __ __|  |     |
  |     |     |     |
  |__ __|__ __|__ __|
   __ __ __ __ __ __ __
  |        |           |
  |        |__ __ __ __|
  |__ __ __|__         |
     |__|__ __|__ __ __|  a(9) = 55
     |__ __ __|  |     |
     |     |     |     |
     |__ __|__ __|__ __|
      __ __ __ __ __ __ __
     |        |           |
     |        |__ __ __ __|
   __|__ __ __|__         |
  |     |__|__ __|__ __ __|  a(10) = 64
  |     |__ __ __|  |     |
  |     |     |     |     |
  |     |__ __|__ __|__ __|
  |__ __|
.

Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2015.

Cf. A347581, A001168, A291808, A291809, A328020, A291806, A006983.

cp's OEIS Frontend

A291806 The number of polyomino tilings of n X n square.

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A291808 Number of tilings of an n X n square using distinct polyominoes.

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A291807 The number of symmetric polyomino tilings of n X n square.

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A347581 The Barnyard sequence: a(n) is the minimum number of unit length line segments required to enclose areas of 1 through n on a square grid.

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A348149 Variation of the Barnyard sequence A347581: a(n) is the minimum number of unit-length line segments required to enclose areas of 1 through n on a square grid when the number of segments is minimized as each area of incrementing size, starting at 1, is added.

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