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

Author

N. J. A. Sloane

Keywords

• sign
• more

Comments

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

References

• 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).

Links

• 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.

Crossrefs

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

Rows 5, 8, and 9 of A383735.

Extensions

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