A342243 -id:A342243 - cp's OEIS frontend

A057730 Number of polyominoes (A000105) with perimeter 2n.

0, 1, 1, 3, 6, 25, 86, 416, 1988, 10640, 57987, 328956, 1900321, 11204350, 67042778, 406780346

Offset: 1

N. J. A. Sloane, Oct 29 2000

Does this include polyominoes with holes? - Franklin T. Adams-Watters, Sep 12 2006. Answer from R. J. Mathar: Yes! See the illustrations in the links (e.g. perimeter 16, area 7, No 81 or perimeter 16, area 8, No 174).

All lines (sides of cells which are not common to a pair of cells) contribute to the perimeter, including the interior sides of cavities and holes. - R. J. Mathar, Feb 19 2021

Andrew Clarke, Isoperimetrical Polyominoes
Andrew Clarke, Picture from Andrew Clarke's page showing the polyominoes of perimeters 4, 6, 8 and 10.
Gordon Hamilton, Grade 3 area and perimeter activity
R. J. Mathar, perimeter 10
R. J. Mathar, perimeter 12
R. J. Mathar, perimeter 14
R. J. Mathar, Illustrating perimeter 16, area <= 13
R. J. Mathar, Illustrating perimeter 18, area <= 13
R. J. Mathar, Illustrating perimeter 20, area <= 13

Cf. A000105, A002931, A057753, A266549 (same, but holes not allowed), column sums of A342243, A131487 (polyominoes by total number of edges).

Additional comments from Barry Cipra, Jun 08 2004
Link updated by William Rex Marshall, Dec 16 2009
a(9)-a(10) added by Luca Petrone, Jan 08 2016
a(1)-a(9) confirmed by Bert Dobbelaere, Oct 19 2018
a(10)-a(12) corrected and extended by John Mason, Jul 26 2021
a(13)-a(16) added by John Mason, Sep 08 2022

A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 83, 255, 847, 2829, 9734, 33724, 118245, 416816, 1478602, 5267171, 18840144, 67611472, 243378415, 878407170, 3178068821, 11523323634, 41865833602, 152382134767

Offset: 1

Tanya Khovanova, Jul 27 2007

2n+2 is the maximal perimeter of an n-celled polyomino. a(n) is the number of n-celled polyominoes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge).

Andrew Clarke, Isoperimetrical Polyominoes, The Poly Pages.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).

Cf. A000105, A057730. Diagonal of A342243.
A359522 counts only polyominoes with holes.
A002013 counts only unbranched polyominoes.
A038142 is the analog for polyhexes.

a(n) <= A000105(n), a(n) <= A057730(n+1).

a(n) >= A000602(n) [see comment on edge graph trees]. - R. J. Mathar, Mar 08 2021

a(14)-a(16) from David Radcliffe, Dec 25 2017
a(17) from David Radcliffe, Dec 26 2017
a(18)-a(24) from John Mason, Dec 11 2021

A100092 Number of n-celled polyominoes with minimum perimeter.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 6, 1, 1, 11, 4, 2, 1, 11, 6, 1, 1, 28, 11, 4, 2, 1, 35, 11, 6, 1, 1, 65, 28, 11, 4, 2, 1, 73, 35, 11, 6, 1, 1, 147, 65, 28, 11, 4, 2, 1, 182, 73, 35, 11, 6, 1, 1, 321, 147, 65, 28, 11, 4, 2, 1, 374, 182, 73, 35, 11, 6, 1, 1, 678, 321, 147, 65, 28, 11, 4, 2, 1

Offset: 0

Sascha Kurz, Nov 03 2004

nonn

An n-celled polyomino has minimum perimeter A027709(n) = 2*ceiling(2*sqrt(n)). - Dmitry Kamenetsky, Feb 27 2017

			a(9) = 1 because the 3 X 3 square is the unique polyomino with minimum perimeter.

Joerg Arndt, Table of n, a(n) for n = 0..144
Sascha Kurz, Counting polyominoes with minimum perimeter, submitted to Ars Combinatoria
Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2005-2015.
Kival Ngaokrajang, Illustration of initial terms ["A275966" should be changed to "A100092"]
Piotr Pikul, Tightest Arrangements of All Different Nets of a Cube, Math. Mag. (2024).

Cf. A027709, A100093, A100094, left nonzero term in row n of A342243.

(* Warning: some local maxima are precomputed from A100094. *)
A100094 = {2, 4, 11, 28, 65, 147, 321, 678, 1382, 2738, 5289 (* extend if needed *)};
amax = Last[A100094]; nmax = 144;
S[x_] := 1 + Sum[ x^(2*n + 1)*Product[ (x^(2*k - 1) - 1), {k, n}], {n, 0, nmax}] + O[x]^nmax;
A[x_] = Product[1/(1 - x^k), {k, 1, nmax}] + O[x]^nmax // Normal;
R[x_] := 1/4 (A[x]^4 + 3A[x^2]^2) + O[x]^nmax;
Q[x_] := 1/8 (A[x]^4 + 3A[x^2]^2 + 2S[x]^2 A[x^2] + 2A[x^4]) + O[x]^nmax;
r[k_] := SeriesCoefficient[R[x], {x, 0, k}];
q[k_] := SeriesCoefficient[Q[x], {x, 0, k}];
e[n_] := Module[{s, w}, s = Floor[Sqrt[n]]; a94Q[k_] := IntegerQ[w = Sqrt[k + n] - k] && w > 0; Which[Evaluate[Sequence @@ Flatten[Table[{a94Q[k], A100094[[k]]}, {k, 3, Length[A100094]}]]], n == s^2, 1, IntegerQ[t = n - s^2] && 0 < t < s, Sum[r[s - c - c^2 - t], {c, 0, Floor[-1/2 + (1/2)* Sqrt[1 + 4 s - 4 t]]}], n == s^2 + s, 1, IntegerQ[t = n - s^2 - s] && 0 < t <= s, q[s + 1 - t] + Sum[r[s + 1 - c^2 - t], {c, 1, Floor[Sqrt[s + 1 - t]]}], True, Print["error n = ", n]]];
Select[Table[e[n], {n, 0, nmax}], # <= amax&] (* Jean-François Alcover, Jul 20 2018 *)

It seems that for m >= 1, 0 <= k <= m-1, we have a(m^2-k) = a(k^2+k+1) = A100094(k) and a(m^2+m-k) = a((k+1)^2+1) = A100093(k+1). If this is true, then a(n) = 1 if and only if n is of the form m^2, m^2 + m - 1 or m^2 + m. - Jianing Song, Aug 10 2021

Offset changed to 0 by N. J. A. Sloane, Mar 19 2017

A131487 a(n) is the number of polyominoes with n edges, including inner edges.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 4, 0, 1, 11, 1, 7, 27, 4, 21, 85, 21, 92, 264, 89, 345, 914, 394, 1405, 3155, 1736, 5530, 11400, 7586, 22022, 41756, 32702, 87158, 156412, 139253, 346836, 592661, 589101, 1379837, 2275935, 2476770, 5501846, 8830267, 10363627, 21970992, 34594887, 43188260, 87950618

Offset: 1

Tanya Khovanova, Jul 28 2007

An n-celled polyomino with perimeter p has (4n+p)/2 edges. The maximum number of edges in an n-celled polyomino is 3n+1.

			A single cell has 4 edges; a domino has 7 edges (this includes the edge between the two cells); both trominoes have 10 edges; their possible orientations are not considered distinct. Thus a(4) = a(7) = 1, a(10) = 2, and a(n) = 0 for n < 10 not equal to 4 or 7.
a(22) = 85 = 83 + 2: there are 83 polyominoes with 7 cells and perimeter 16 (such as a 1 X 7 strip) and two polyominoes with 8 cells and perimeter 12 (a 3 X 3 square without a corner and a 4 X 2 rectangle), and each of these polyominoes has 22 edges.
a(23) = 21. a(24) = 91+1. a(25) = 255+9. a(26) = 89. a(27) = 339+6. a(28) = 847+67. a(34) = 9734+1655+11. a(35) = 7412+174. - _R. J. Mathar_, Feb 22 2021

Andrew Clarke, Isoperimetrical Polyominoes

Cf. A000105, A057730, A342243.

Cf. A131482 (number of n-celled polyominoes with perimeter 2n+2), A131488 (analog for hexagonal tiling).

See A342243 for formula.

a(23)-a(35) from R. J. Mathar, Feb 22 2021
a(36)-a(39) from R. J. Mathar, Mar 11 2021
a(40)-a(44) from R. J. Mathar, Mar 24 2021
a(45)-a(54) from John Mason, Apr 28 2023

A380287 Sum of the perimeters of the free polyominoes with n cells.

Original entry on oeis.org

4, 6, 16, 48, 142, 472, 1670, 6364, 24604, 97668, 390070, 1570560, 6334644, 25617062, 103669288, 419930444, 1701635046, 6898183050

Offset: 1

Omar E. Pol, Jan 25 2025

The perimeters of any holes are included here.

			Illustration for n = 4:
The free polyominoes with four cells are also called free tetrominoes.
The five free tetrominoes are as shown below:
    _
   |_|     _       _       _
   |_|    |_|     |_|_    |_|_     _ _
   |_|    |_|_    |_|_|   |_|_|   |_|_|
   |_|    |_|_|     |_|   |_|     |_|_|
.
From left to right the perimeters are respectively [10, 10, 10, 10, 8] as shown below:
    _
   | |     _       _       _
   | |    | |     | |_    | |_     _ _
   | |    | |_    |_  |   |  _|   |   |
   |_|    |_ _|     |_|   |_|     |_ _|
.
The sum of the perimeters is 10 + 10 + 10 + 10 + 8 = 48, so a(4) = 48.
.

Index entries for sequences related to polyominoes.

Cf. A000105, A027709, A057730, A130622, A131482, A135942, A342243.

See A380575 for another version.

a(n) = Sum_{k=2..n+1} 2*k*A342243(n,k). - Pontus von Brömssen, Jan 27 2025

a(6)-a(18) (using A342243 b-file) from Pontus von Brömssen, Jan 27 2025

A380575 One half of the sum of the perimeters of the free polyominoes with n cells.

Original entry on oeis.org

2, 3, 8, 24, 71, 236, 835, 3182, 12302, 48834, 195035, 785280, 3167322, 12808531, 51834644, 209965222, 850817523, 3449091525

Offset: 1

Omar E. Pol, Feb 12 2025

A380287 is the main entry for this question.

One half of the perimeters of any holes are included here.

Index entries for sequences related to polyominoes.

Cf. A000105, A342243, A380287.

a(n) = A380287(n)/2.

a(n) = Sum_{k=2..n+1} k*A342243(n,k). - Andrew Howroyd, Feb 28 2025

cp's OEIS Frontend

A057730 Number of polyominoes (A000105) with perimeter 2n.

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A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

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A100092 Number of n-celled polyominoes with minimum perimeter.

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A131487 a(n) is the number of polyominoes with n edges, including inner edges.

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A380287 Sum of the perimeters of the free polyominoes with n cells.

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A380575 One half of the sum of the perimeters of the free polyominoes with n cells.

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