This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003203 M3433 #37 May 21 2025 09:21:04
%S A003203 1,4,12,24,52,108,224,412,844,1528,3152,5036,11984,15040,46512,34788,
%T A003203 197612,4036,929368,-702592,4847552,-7033956,27903296,-54403996,
%U A003203 170579740
%N A003203 Cluster series for square lattice.
%C A003203 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 A003203 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 A003203 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003203 John Adler, <a href="https://doi.org/10.1063/1.168493">Series Expansions</a>, Computers in Physics, 8 (1994), 287-295.
%H A003203 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 A003203 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a003/A003203.java">Java program</a> (github).
%H A003203 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 A003203 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.
%H A003203 <a href="/index/Cl#cluster">Index entries for sequences related to cluster series</a>.
%Y A003203 Cf. A001168, A003202 (triangular net), A003204 (honeycomb net), A003198 (bond percolation), A338210 (perimeter polynomials).
%Y A003203 Rows 5, 8, and 9 of A383735.
%K A003203 sign,more
%O A003203 0,2
%A A003203 _N. J. A. Sloane_
%E A003203 a(11)-a(14) from _Sean A. Irvine_, Aug 15 2020
%E A003203 a(15)-a(24) added from Conway & Guttmann by _Andrey Zabolotskiy_, Feb 01 2022