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

Author

Robert P. P. McKone, Oct 10 2023

Keywords

• nonn

Examples

			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}.
		

Links

• 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).

Crossrefs

Cf. A366551 (3 X 3 matrices), A367978 (4 X 4 matrices).

Cf. A058331 (determinants), A005408 (traces).

Cf. A272659, A365926.

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

PARI

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

Python

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

Formula

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