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.
a(4) = 9 because the permanents of non-singular 4 X 4 (0,1)-matrices can take the values 1,2,..,7,9,11.
The 3 X 3 matrix ((0,1,0),(0,0,1),(1,1,1)) has real eigenvalue 1.83929 and the complex pair -0.41964+-0.60629*i. There are 12 (0,1) 3 X 3 matrices with these eigenvalues. There are 6 groups of 6 matrices having eigenvalues (1.3472,-0.66236+-0.56228*i), (1.46557,-0.23279+-0.79255*i),..., (2.32472,0.33764+-0.56228*i). Two matrices (e.g. ((0,0,1),(1,0,0),(0,1,0)) ) have eigenvalues (1,-0.5+-0.5*sqrt(3)*i). Two matrices (e.g. ((1,1,0),(0,1,1),(1,0,1)) ) have eigenvalues (2,0.5+-0.5*sqrt(3)*i). Total: 12+6*6+2+2=52=a(3).
a[n_] := Module[{M, iter, cnt=0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[AnyTrue[Eigenvalues[M], Im[#] != 0&], cnt++], Evaluate[Sequence @@ iter]]; cnt];
Do[Print[n, " ", a[n]], {n, 1, 4}] (* Jean-François Alcover, Dec 09 2018 *)
From _M. F. Hasler_, May 25 2020: (Start) Representatives of the two inequivalent nonsingular (0,1) matrices for n=2 are [ 1 0 ] and [ 1 1 ] . [ 0 1 ] [ 0 1 ] For n=3 we have 8 nonsingular nonequivalent representatives: [1 0 0] [1 0 0] [1 0 1] [1 1 1] [1 1 0] [1 1 0] [1 1 1] [1 1 0] [0 1 0], [0 1 1], [0 1 1], [0 1 0], [0 1 1], [1 0 1], [0 1 1], [1 0 1]. [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 0 1] [1 1 1] To see that they are inequivalent, consider their column sums: (1 1 1), (1 1 2), (1 1 3), (1 2 2), (1 2 2), (2 2 2), (1 2 3), (3 2 2). Only the 4th and 5th matrix have equivalent column sum signature (1,2,2), but their row sums are (3,1,1) resp. (2,2,1). Therefore they can't be obtained one from the other by row and column permutations which leave invariant these sums. (End)
(* 2x2 case *) cnt = 0; Do[d = Det[{{a, b}, {c, d}}]; If[d != 0, cnt++], {a, 0, 2}, {b, 0, 2}, {c, 0, 2}, {d, 0, 2}]; cnt (* T. D. Noe, Nov 29 2011 *)
For n = 2 the six matrices are (+ means +1, - means -1): ++ +- -- -+ +- ++ -- +- ++ -+ -+ ++ with eigenvalues 00 00 00 00 20 20 respectively.
a[n_] := Select[Partition[#, n] & /@ Tuples[{-1, 1}, {n^2}], AllTrue[ Eigenvalues[#], NonNegative]&] // Length; a[0] = 1;
Do[Print[n, " ", a[n]], {n, 0, 5}] (* Jean-François Alcover, Feb 13 2019 *)
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