A366443 -id:A366443 - cp's OEIS frontend

A338211 Triangle of coefficients of perimeter polynomials for free polyominoes.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 10, 10, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 18, 37, 30, 15, 3, 1

Offset: 0

Sean A. Irvine, Oct 17 2020

Considered as a triangle, T(n,k) is the number of free polyominoes of n cells having a (cell) perimeter of k.

			Polynomials begin:
  1;
  x^4;
  x^6;
  x^7 + x^8;
  3*x^8 + x^9 + x^10;
  ...

Sean A. Irvine, Table of n, a(n) for n = 0..437; rows 0..19 flattened
Sean A. Irvine, Java program (github)

Cf. A000105 (row sums), A338210 (fixed equivalent), A338213 (sprawl), A366443 (column sums).

A000105(n) = Sum_{k=0..2*n+2} T(n,k).

A216820 Number of polyominoes of site-perimeter n with 8-holes allowed.

Original entry on oeis.org

1, 0, 2, 4, 12, 32, 110, 340, 1209, 4272, 16166, 61848, 246660, 1004883, 4209124, 18020832, 78898047, 352437205, 1605225878, 7445515638, 35142033027, 168644213617, 822311934788, 4071431204506, 20457850555113

Offset: 4

N. J. A. Sloane, Sep 20 2012

This sequence counts fixed connected (via common edges) polyominoes with given site-perimeter. The site-perimeter of a polyomino is the number of cells that are adjacent to it (via common edges). This sequence allows holes of any kind; A216819 allows holes but requires them to be connected to each other and to the exterior area via common corners; A216818 doesn't allow holes. - Andrey Zabolotskiy, Feb 02 2022

			The only polyomino with site-perimeter 4 is a single cell.
No polyominoes have site-perimeter 5.
a(6) = 2: the domino, rotated (or reflected) in 2 possible ways.
a(7) = 4: the L-tromino, rotated in 4 ways.
a(8) = 12: the X-pentomino; the square tetromino; the straight tromino, rotated in 2 ways; the T-tetromino, rotated in 4 ways; the skew tetromino, rotated and reflected in 4 ways.

A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen., 28 (1995), 891-904. See Table 2.
J. Fortier, A. Goupil, J. Lortie and J. Tremblay, Exhaustive generation of gominoes, Theoretical Computer Science, 502 (2013), 76-87. See Table 1, beware of the typo in a(15).

Cf. A216818 (no holes), A216819 (holes connected by corners); A001168 (by area), A057730 (by perimeter); A366443 (free).

a(15) corrected, a(16)-a(28) from Conway & Guttmann added by Andrey Zabolotskiy, Feb 02 2022

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A338211 Triangle of coefficients of perimeter polynomials for free polyominoes.

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