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

0,2

Links

Robert P. P. McKone, Table of n, a(n) for n = 0..149

Robert P. P. McKone, The distinct characteristic polynomials for a(0)-a(23).

Formula

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

Example

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}.

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}]

Prog

(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

Crossrefs

Cf. A366551 (3 X 3 matrices).

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

Cf. A272659, A365926.

Sequence in context: A081441 A363614 A363606 * A127760 A320243 A066188

Adjacent sequences: A366445 A366446 A366447 * A366449 A366450 A366451

Keyword

nonn

Author

Robert P. P. McKone, Oct 10 2023

Status

approved