A219736 The number of 2 X 2 matrices with all eigenvalues real and whose entries are integers with absolute value at most n.

67, 477, 1759, 4729, 10395, 20141, 35423, 58225, 90579, 134813, 193503, 269785, 366267, 486925, 635199, 815105, 1030371, 1286221, 1586447, 1937033, 2342379, 2808221, 3340239, 3945361, 4628467, 5396781, 6257039, 7216457, 8281579, 9461805, 10762495, 12193873

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, A207259.

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 - A207259(n).

Extensions

a(16)-a(32) from Hiroaki Yamanouchi, Oct 03 2014