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%I #9 Nov 26 2023 15:08:08
%S 1,32,513,4407,21393,86620,242057,673623,1467642,3107487,5836467,
%T 11108595
%N Number of distinct characteristic polynomials for 3 X 3 matrices with entries from {0, 1, ..., n}.
%H Robert P. P. McKone, <a href="/A366551/a366551_1.txt">The distinct characteristic polynomials for a(0)-a(6)</a>.
%F a(n) <= A366158(n) * A227776(n) * A016777(n).
%t mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}]; flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]]; charPolyMat[n_Integer?Positive] := charPolyMat[n] = FullSimplify[CoefficientList[Expand[CharacteristicPolynomial[mat[n], x]], x]]; a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[Table[Evaluate[charPolyMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}], d^2 - 1]]]; Table[a[3, n], {n, 0, 7}]
%Y Cf. A366448 (2 X 2 matrices).
%Y Cf. A366158 (determinants), A227776 (2nd order coefficients), A016777 (traces).
%Y Cf. A272659.
%K nonn,more
%O 0,2
%A _Robert P. P. McKone_, Oct 13 2023