A207259 The number of 2 X 2 matrices with no real eigenvalues and whose entries are integers of absolute value at most n.

14, 148, 642, 1832, 4246, 8420, 15202, 25296, 39742, 59668, 86338, 120840, 165174, 220356, 288322, 370816, 470254, 587940, 726994, 888728, 1076422, 1292404, 1539442, 1819440, 2136734, 2493700, 2893586, 3339544, 3835782, 4384036, 4990466, 5656752, 6388158

Offset: 1

Author

W. Edwin Clark, Nov 26 2012

Keywords

• nonn
• easy

Links

• Hiroaki Yamanouchi and Chai Wah Wu, Table of n, a(n) for n = 1..1000 (terms for n = 1..100 from Hiroaki Yamanouchi)

Crossrefs

Cf. A003024, A219736, A219744.

Programs

Maple

a:=proc(n)
local x,y,z,w,Eig,count;
count:=0;
for x from -n to n do
for y from -n to n do
for z from -n to n do
for w from -n to n do
Eig:=LinearAlgebra:-Eigenvalues(Matrix([[x,y],[z,w]]));
if Im(Eig[1]) <> 0  then count:=count+1; fi;
od:
od:
od:
od:
count;
end:

Formula

a(n) = (2*n+1)^4 - A219736(n).

Extensions

a(16)-a(33) from Hiroaki Yamanouchi, Oct 03 2014