A003203 - OEIS

%I M3433 #33 Feb 02 2022 15:59:09

%S 1,4,12,24,52,108,224,412,844,1528,3152,5036,11984,15040,46512,34788,

%T 197612,4036,929368,-702592,4847552,-7033956,27903296,-54403996,

%U 170579740

%N Cluster series for square lattice.

%C 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

%D 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.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H John Adler, <a href="https://doi.org/10.1063/1.168493">Series Expansions</a>, Computers in Physics, 8 (1994), 287-295.

%H A. R. Conway and A. J. Guttmann, <a href="https://doi.org/10.1088/0305-4470/28/4/015">On two-dimensional percolation</a>, J. Phys. A: Math. Gen., 28 (1995), 891-904. See Table 3.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a003/A003203.java">Java program</a> (github)

%H M. F. Sykes and J. W. Essam, <a href="https://doi.org/10.1103/PhysRev.133.A310">Critical percolation probabilities by series methods</a>, Phys. Rev., 133 (1964), A310-A315.

%H M. F. Sykes and M. Glen, <a href="https://doi.org/10.1088/0305-4470/9/1/014">Percolation processes in two dimensions. I. Low-density series expansions</a>, J. Phys. A: Math. Gen., 9 (1976), 87-95.

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

%K sign,more

%O 0,2

%A _N. J. A. Sloane_

%E a(11)-a(14) from _Sean A. Irvine_, Aug 15 2020

%E a(15)-a(24) added from Conway & Guttmann by _Andrey Zabolotskiy_, Feb 01 2022