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:=[0, 0, 0, 0, 3, 9, 16]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..55]];
Join[{0, 0, 0, 0}, Table[(n^2+3n-22)/2, {n, 4, 54}]]
my(x='x + O('x^55)); concat([0, 0, 0, 0], Vec(serlaplace(11 + 9*x + 3*x^2 + x^3/3 + exp(x)*(x^2 + 4*x - 22)/2)))
(x^4*(3 - 2*x^2)/(1 - x)^3).series(x, 55).coefficients(x, sparse=False)
All graphs with at most three nodes are C_4-free, so a(n) = 1 for n <= 3 and any graph is optimal.
For 4 <= n <= 9, the following are all optimal graphs, i.e., graphs that have n nodes and a(n) maximal C_4-free subgraphs:
n = 4: the 4-cycle;
n = 5: K_{2,3};
n = 6: the prism graph and the octahedral graph;
n = 7: the complement of 2*K_2 + K_3;
n = 8: K_4 X K_2 (Cartesian product) and the 16-cell;
n = 9: the circulant graph C_9(1,3), and K_{3,3,3} with three edges removed, one edge between the first and second parts in the partition and two edges from two other nodes in these two parts to a node in the third part.
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