A366551 Number of distinct characteristic polynomials for 3 X 3 matrices with entries from {0, 1, ..., n}.

1, 32, 513, 4407, 21393, 86620, 242057, 673623, 1467642, 3107487, 5836467, 11108595, 18102935, 31327359, 48505904, 74802671, 110297111, 166721570, 230270840

Offset: 0

Author

Robert P. P. McKone, Oct 13 2023

Keywords

• nonn
• more

Links

• Robert P. P. McKone, The distinct characteristic polynomials for a(0)-a(6).

Crossrefs

Cf. A366448 (2 X 2 matrices), A367978 (4 X 4 matrices).

Cf. A366158 (determinants), A227776 (2nd order coefficients), A016777 (traces).

Cf. A272659.

Programs

Mathematica

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}]

Sage

import itertools
def a(n):
    ans, W = set(), itertools.product(range(n+1), repeat=9)
    for w in W: ans.add(Matrix(ZZ, 3, 3, w).charpoly())
    return len(ans)  # Robin Visser, May 08 2025

Formula

a(n) <= A366158(n) * A227776(n) * A016777(n).

Extensions

a(12)-a(18) from Robin Visser, May 08 2025