A366158 Number of distinct determinants of 3 X 3 matrices with entries from {0, 1, ..., n}.

1, 5, 25, 77, 179, 355, 609, 995, 1497, 2167, 2999, 4069, 5289, 6841, 8595, 10661, 13023, 15777, 18795, 22305, 26085, 30397, 35107, 40381, 45929, 52247, 58929, 66287, 74139, 82767, 91643, 101701, 112013, 123235

Offset: 0

Author

Robert P. P. McKone, Oct 02 2023

Keywords

• nonn
• more
• changed

Comments

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.

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

Links

• Robert P. P. McKone, The distinct determinants for a(0)-a(18).

Crossrefs

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

Cf. A272661, A272659.

Cf. A365926.

Cf. A272658, A272660, A272662, A272663.

Cf. A033431 (maximal determinant).

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

Programs

Mathematica

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}]

Python

from itertools import product
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

Extensions

a(19)-a(26) from Robin Visser, May 08 2025

a(27)-a(33) from Robin Visser, Aug 26 2025