A338211 -id:A338211 - cp's OEIS frontend

A338210 Triangle of coefficients of perimeter polynomials for fixed polyominoes.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 8, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 12, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 54, 80, 60, 16, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 136, 252, 228, 100, 20, 2

Offset: 0

Sean A. Irvine, Oct 16 2020

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

			Polynomials begin:
  1;
  x^4;
  2*x^6;
  4*x^7 + 2*x^8;
  9*x^8 + 8*x^9 + 2*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. A001168 (row sums), A338211 (free equivalent), A338212 (sprawl), A003203.

A001168(n) = Sum_{k=0..2*n+2} T(n,k).

A338213 a(n) is the number of free polyominoes with sprawl n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 3, 2, 4, 6, 13, 15, 30, 50, 93, 158, 287, 513, 950, 1735, 3212, 5945, 11099, 20720, 38910, 73002, 137821, 259757

Offset: 0

Sean A. Irvine and Allan C. Wechsler, Oct 17 2020

The sprawl of a polyomino is the number of cells in a polyomino plus the number of cells adjacent to it.

Sean A. Irvine, Java program (github)

Cf. A000105, A338211, A338212.

Antidiagonal sums of A338211.

A366443 Number of free polyominoes of site-perimeter n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 5, 5, 23, 46, 187, 552, 2145, 7818

Offset: 1

John Mason from an idea of Allan C. Wechsler, Oct 10 2023

This sequence counts free 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.

			a(4) = a(6) = a(7) = 1 as the monomino, domino and L-shaped tromino are the only polyominoes with site perimeter 4, 6 and 7 respectively.
a(5) = 0 as no polyomino has a site-perimeter of 5.
a(8) = 5 as the straight tromino, square tetromino, T-tetromino, S-tetromino and cross pentomino are the only polyominoes with site perimeter 8. See link "Examples".

John Mason, Examples

Cf. A000105 (free polyominoes), A001971 (the maximum size of a polyomino with site-perimeter n is given by A001971(n-2)), A057730 (perimeter instead of site-perimeter), A216820 (fixed version of current sequence).

Column sums of A338211 (without the column for 0-celled polyominoes).

a(15) corrected by Sean A. Irvine, Apr 13 2025

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A338210 Triangle of coefficients of perimeter polynomials for fixed polyominoes.

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A338213 a(n) is the number of free polyominoes with sprawl n.

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A366443 Number of free polyominoes of site-perimeter n.

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