A365926 -id:A365926 - cp's OEIS frontend

A366448 Number of distinct characteristic polynomials for 2 X 2 matrices with entries from {0, 1, ..., n}.

1, 6, 22, 58, 116, 221, 356, 573, 824, 1163, 1565, 2143, 2697, 3527, 4385, 5388, 6455, 7992, 9342, 11262, 12953, 15034, 17301, 20246, 22595, 25823, 29054, 32679, 36228, 41112, 44964, 50600, 55288, 60770, 66543, 72927, 78173, 86577, 93925, 101775, 108798

Offset: 0

Robert P. P. McKone, Oct 10 2023

nonn

			For n = 1 the a(1) = 6 characteristic polynomials are {x^2, -4 + x^2, -2 + x^2, -1 + x^2, -4*x + x^2, 2-4*x + x^2}.

Robin Visser, Table of n, a(n) for n = 0..1000 (terms n = 0..149 from Robert P. P. McKone).
Robert P. P. McKone, The distinct characteristic polynomials for a(0)-a(23).

Cf. A366551 (3 X 3 matrices), A367978 (4 X 4 matrices).
Cf. A058331 (determinants), A005408 (traces).
Cf. A272659, A365926.

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}],3]]]; Table[a[2,n],{n,0,41}]

a(n) = my(list=List()); for (i=0, n, for (j=0, n, for(k=0, n, for(m=0, n, my(p=charpoly([i,j;k,m])); listput(list, p))))); #Set(list); \\ Michel Marcus, Oct 11 2023

def A366448(n): return len({(a+d,a*d-b*c) for a in range(n+1) for b in range(n+1) for c in range(b+1) for d in range(a+1)}) # Chai Wah Wu, Oct 12 2023

a(n) <= A058331(n) * A005408(n) = 4*n^3 + 2*n^2 + 2*n + 1.

A058331(n) = 2*n^2 + 1 <= a(n). - Charles R Greathouse IV, May 08 2025

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

Original entry on oeis.org

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

Robert P. P. McKone, Oct 02 2023

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.

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

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).

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}]

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

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

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

cp's OEIS Frontend

A366448 Number of distinct characteristic polynomials for 2 X 2 matrices with entries from {0, 1, ..., n}.

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