A338413 Number of 2 X 2 matrices with integer entries in [-n,n] that are diagonalizable over the complex numbers.

65, 569, 2281, 6313, 14265, 28033, 49921, 82545, 128945, 192809, 277849, 388185, 528617, 704049, 919857, 1181393, 1495569, 1868249, 2306921, 2818441, 3410809, 4091937, 4870273, 5754449, 6753233, 7877641, 9136441, 10540633, 12101001, 13828465, 15734545, 17830353, 20129713, 22644553, 25387929

Offset: 1

Author

Matthew Niemiro, Nov 07 2020

Keywords

• nonn

Comments

A diagonalizable matrix A is one which can be expressed as XDY, where D is a diagonal matrix and X = Y^-1 are square matrices. By 'diagonalizable over C,' it is meant that the matrix D has complex entries.

The nondiagonalizable 2 x 2 matrices are the nondiagonal ones whose characteristic polynomial has discriminant 0. - Robert Israel, Nov 12 2020

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Diagonalizable matrix

Crossrefs

a(1) is given by A091470(2).

Programs

Maple

N:= 30: # for a(1)..a(N)
V:= Vector(N):
for t from 1 to N do
  for d in select(`<=`,numtheory:-divisors(t^2),N) do
    for n from max(d, t^2/d) to N do
      V[n]:= V[n] + (8*(n-t)+4)
od od od:
for n from 1 to N do V[n]:= (2*n+1)^4 - (V[n] + 4*n*(2*n+1)) od:
convert(V,list); # Robert Israel, Nov 12 2020

Mathematica

a[n_] := Length[Select[Tuples[Tuples[Range[-n, n], 2], 2], DiagonalizableMatrixQ]];

Extensions

More terms from Robert Israel, Nov 12 2020