A338210 -id:A338210 - cp's OEIS frontend

A003203 Cluster series for square lattice.

1, 4, 12, 24, 52, 108, 224, 412, 844, 1528, 3152, 5036, 11984, 15040, 46512, 34788, 197612, 4036, 929368, -702592, 4847552, -7033956, 27903296, -54403996, 170579740

Offset: 0

N. J. A. Sloane

The word "cluster" here essentially means polyomino or animal. This sequence can be computed based on a calculation of the perimeter polynomials of polyominoes. In particular, if P_n(x) is the perimeter polynomial for all fixed polyominoes of size n, then this sequence is the coefficients of x in Sum_{k>=1} k^2 * x^k * P_k(1-x). - Sean A. Irvine, Aug 15 2020

J. W. Essam, Percolation and cluster size, in C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Ac. Press 1972, Vol. 2; see especially pp. 225-226.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Adler, Series Expansions, Computers in Physics, 8 (1994), 287-295.
A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen., 28 (1995), 891-904. See Table 3.
Sean A. Irvine, Java program (github).
M. F. Sykes and J. W. Essam, Critical percolation probabilities by series methods, Phys. Rev., 133 (1964), A310-A315.
M. F. Sykes and M. Glen, Percolation processes in two dimensions. I. Low-density series expansions, J. Phys. A: Math. Gen., 9 (1976), 87-95.
Index entries for sequences related to cluster series.

Cf. A001168, A003202 (triangular net), A003204 (honeycomb net), A003198 (bond percolation), A338210 (perimeter polynomials).

Rows 5, 8, and 9 of A383735.

a(11)-a(14) from Sean A. Irvine, Aug 15 2020

a(15)-a(24) added from Conway & Guttmann by Andrey Zabolotskiy, Feb 01 2022

A338211 Triangle of coefficients of perimeter polynomials for free polyominoes.

Original entry on oeis.org

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).

A338212 a(n) is the number of fixed polyominoes with sprawl n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 4, 2, 9, 9, 22, 32, 66, 104, 200, 348, 646, 1172, 2144, 3964, 7285, 13581, 25136, 47070, 87700, 164764, 309308, 582152, 1098698, 2074484

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. A001168, A338210, A338213.

Antidiagonal sums of A338210.

cp's OEIS Frontend

A003203 Cluster series for square lattice.

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A338211 Triangle of coefficients of perimeter polynomials for free polyominoes.

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A338212 a(n) is the number of fixed polyominoes with sprawl n.

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