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Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.
2

%I #38 Oct 23 2023 10:38:36

%S 1,5,25,77,179,355,609,995,1497,2167,2999,4069,5289,6841,8595,10661,

%T 13023,15777,18795

%N Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

%C These determinants a(n) equivalently represent the leading coefficient (coefficient of term with degree 0) of the characteristic polynomials for such matrices, thereby providing a direct measure and lower bound of the uniqueness of these polynomials within this matrix class.

%C The maximal determinant counted by a(n) is A033431(n) = 2*n^3.

%H Robert P. P. McKone, <a href="/A366158/a366158.txt">The distinct determinants for a(0)-a(18)</a>.

%t mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}]; flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]]; detMat[n_Integer?Positive] := detMat[n] = Det[mat[n]] // FullSimplify; a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[ParallelTable[Evaluate[detMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}]]]]; Table[a[3, n], {n, 0, 9}]

%o (Python)

%o from itertools import product

%o def A366158(n): return len({a[0]*(a[4]*a[8] - a[5]*a[7]) - a[1]*(a[3]*a[8] - a[5]*a[6]) + a[2]*(a[3]*a[7] - a[4]*a[6]) for a in product(range(n+1),repeat=9)}) # _Chai Wah Wu_, Oct 06 2023

%Y Cf. A058331 (distinct determinants for 2 X 2 matrices).

%Y Cf. A272661, A272659.

%Y Cf. A365926.

%Y Cf. A272658, A272660, A272662, A272663.

%Y Cf. A033431 (maximal determinant).

%Y Cf. A097400 (distinct consecutive entries in 3 X 3 matrix).

%K nonn,more

%O 0,2

%A _Robert P. P. McKone_, Oct 02 2023