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

1,1

Links

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

Formula

a(n) = (2*n+1)^4 - A207259(n).

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:

Crossrefs

Cf. A003024, A207259.

Sequence in context: A323577 A142735 A342803 * A257747 A163150 A120717

Adjacent sequences: A219733 A219734 A219735 * A219737 A219738 A219739

Keyword

nonn,easy

Author

W. Edwin Clark, Nov 26 2012

Extensions

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

Status

approved