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 A098148 Number of real (0,1) n X n matrices such that some eigenvalues are strictly complex. 1

%I

%S 0,0,52,22196,21005094

%N Number of real (0,1) n X n matrices such that some eigenvalues are strictly complex.

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%e 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).

%t 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];

%t Do[Print[n, " ", a[n]], {n, 1, 4}] (* _Jean-François Alcover_, Dec 09 2018 *)

%Y Cf. other counts for (0, 1) matrices: A003024 (positive eigenvalues), A055165 (nonsingular), A085656 (positive definite), A086510 (nonnegative eigenvalues).

%K more,nonn

%O 1,3

%A _Hugo Pfoertner_, Sep 07 2004

%E a(5) corrected by _Hugo Pfoertner_, Sep 26 2017

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Last modified July 25 12:05 EDT 2021. Contains 346289 sequences. (Running on oeis4.)