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

1,1

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

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]];

Crossrefs

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

Sequence in context: A222588 A286612 A276963 * A211404 A297317 A224099

Adjacent sequences: A338410 A338411 A338412 * A338414 A338415 A338416

Keyword

nonn

Author

Matthew Niemiro, Nov 07 2020

Extensions

More terms from Robert Israel, Nov 12 2020

Status

approved