A001168 -id:A001168 - cp's OEIS frontend

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.

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.

A353978 Number of fixed polytans (polyaboloes) with n cells.

Original entry on oeis.org

4, 9, 24, 71, 224, 740, 2496, 8565, 29792, 104701, 371304, 1326702, 4771380, 17256161, 62712800, 228883359, 838492436, 3081972336, 11361867384, 41998361480, 155620033360, 577900838281

Offset: 1

Aaron N. Siegel, May 12 2022

Cf. A001168, A006074, A151519.

a(21)-a(22) from Aaron N. Siegel, Jun 07 2022

A113174 Number of fixed 3D piled polyominoes: polycubes with fixed orientation, with no cubes "sitting on air".

Original entry on oeis.org

1, 3, 11, 44, 184, 792, 3484, 15592, 70745, 324561, 1502511, 7007929, 32892778, 155221536, 735915652, 3503270920, 16737092549, 80218277681, 385574074383, 1858059853316, 8974761939239, 43441619693731, 210682920968681

Offset: 1

Franklin T. Adams-Watters, Oct 15 2005

nonn

			For n = 4, there are 4 orientations of the angled tricube excluded: those which set it on a point; this leaves 8 orientations of the angled tricube and 3 of the straight tricube.

Cf. A001168, A001931 (fixed polycubes).

a(n) = sum_{m=1}^n A001168(m)*C(n-1, m-1). If both sequences are shifted left, binomial transform of A001168.

Corrected by Franklin T. Adams-Watters, Oct 25 2006

A127560 Number of fixed r-celled polyominoes with smallest containing rectangle measuring k by m, read in order r=A056556(n)+1, k=A056560(n)+1, m=A056558(n)+1.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 1, 0, 1, 8, 8, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 1, 12, 25, 12, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 18, 44, 18, 0, 1, 16, 50, 50, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 32, 8, 0, 0, 38, 155, 155, 38, 0, 1, 20, 82, 120, 82, 20, 1

Offset: 0

Graeme McRae, Jan 18 2007

nonn

The sum of each triangle, i.e. for a given r the sum of a(n) for all n such that r=A056556(n)+1 is the number of r-celled fixed polyominoes, A001168(r).

			The 5th triangle of the sequence, the number of fixed pentominoes by dimension, is
0,0,0,0,1
0,0,6,12
0,6,25
0,12
1
This indicates, for example, there are 25 fixed pentominos that fit in a 3 X 3 rectangle and 12 fixed pentominos that fit in a 4 X 2 rectangle.

Cf. A000105, A000988, A001168 Indices for reading by triangles given by A056556, A056560, A056558.

A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

0, 2, 42, 937, 26427, 937126, 40290848, 2036152559, 118202398712, 7747410863508, 565695467280668, 45525704815211707, 4002930269942820774, 381750656962679848234, 39244733577786597223238

Offset: 1

David Bevan, Jul 28 2009

nonn

a(n) = A162676(n) - A001168(n)

			a(2)=2: the two rotations of the disconnected domino consisting of two squares connected at a vertex

Cf. A162676, A001168, A006770, A030222

A171579 Number of fixed polyominoes in equilibrium; a fixed polyomino is in equilibrium when its center of mass is vertically aligned with a cell with minimal coordinate.

Original entry on oeis.org

0, 1, 1, 2, 5, 12, 27, 70, 192, 589, 1825, 5870, 18952, 62997, 211014, 718474, 2468741, 8578400

Offset: 1

Mark Elduck (mark.elduck(AT)gmail.com), Dec 12 2009, Dec 20 2009, Dec 21 2009

			For n=3, we have 2 fixed polyominoes in equilibrium: namely the vertical 3-omino and the horizontal 3-omino, thus a(3)=2.

Cf. A001168

A194596 Number of free polyplets with n cells that are not polyominoes.

Original entry on oeis.org

0, 1, 3, 17, 82, 489, 2923, 18401, 116848, 753726, 4898579, 32085696, 211398614, 1400292492, 9318028028, 62259251309, 417496576187

Offset: 1

Jani Melik, Aug 30 2011

			XX...X.X..X.. (the 3 for n=3)
..X...X....X.
............X
a(3) = 5 - 2 = 3
a(4) = 22 - 5 = 17
a(5) = 94 - 12 = 82

Eric Weisstein's World of Mathematics, Polyplet.

Cf. A030222, A000105, A006770 (fixed polyplets), A001168 (fixed polyominoes).

a(n) = A030222(n) - A000105(n).

A202015 Number of fixed polyominoes that can produce a repeating phenotype with 1, 2, or 4 90-degree turns.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 0, 2, 6, 1, 7, 19, 1, 7, 63, 0, 16, 216, 0, 16, 760, 3, 49, 2725, 2, 48, 9910, 0, 158, 36446

Offset: 1

John Michael Feuk, Dec 08 2011

P is three numbers, according to 90-degree turns of a given polyomino of n squares. Each of the three numbers corresponds to a number of 90-degree turns (1, 2, and 4). Given P=(1), 3 numbers: a(1), a(2), and a(3) can be created. P=(1) refers to (1) squares in a polyomino. a(1) would be the number of 1-square polyominoes that can turn once 90 degrees and still be considered the same phenotypic shape. a(2) would be the number of 1-square polyominoes that can turn twice 90 degrees (180 degrees) and still be considered the same phenotypic shape. a(3) would be the number of 1-square polyominoes that can turn four times 90 degrees (360 degrees) and still be considered the same phenotypic shape. In other words, a(3) is the number of 1-square polyominoes that are not radially symmetric with respect to the y- and x-axes. Now, start over, and given P=(2), 3 numbers: a(4), a(5), and a(6) can be created.

			For P=(1), a(1) = 1, a(2) = 1, and a(3) = 1.
For P=(2), a(4) = 0, a(5) = 2, and a(6) = 2.

Graeme McRae, Polyominoes
Wikipedia, Heptomino Symmetry

Cf. A001168 (use square animals from this list).

A275943 Number of prefix-closed polyominoes of area n.

Original entry on oeis.org

1, 2, 6, 19, 61, 197, 640, 2091, 6860, 22566, 74365, 245397, 810639, 2680025, 8865866

Offset: 1

Enrico Formenti, Aug 13 2016

Enrico Formenti and Paolo Massazza, From Tetris to Polyominoes Generation, Proceedings of GASCom 2016 conference, 2016.

Cf. A000105, A001168.

cp's OEIS Frontend

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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A353978 Number of fixed polytans (polyaboloes) with n cells.

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A113174 Number of fixed 3D piled polyominoes: polycubes with fixed orientation, with no cubes "sitting on air".

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A127560 Number of fixed r-celled polyominoes with smallest containing rectangle measuring k by m, read in order r=A056556(n)+1, k=A056560(n)+1, m=A056558(n)+1.

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A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

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A171579 Number of fixed polyominoes in equilibrium; a fixed polyomino is in equilibrium when its center of mass is vertically aligned with a cell with minimal coordinate.

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A194596 Number of free polyplets with n cells that are not polyominoes.

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A202015 Number of fixed polyominoes that can produce a repeating phenotype with 1, 2, or 4 90-degree turns.

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A275943 Number of prefix-closed polyominoes of area n.

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