Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
A207259
The number of 2 X 2 matrices with no real eigenvalues and whose entries are integers of absolute value at most n.
3
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
1,1
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 - A219736(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
KEYWORD
nonn,easy
AUTHOR
W. Edwin Clark, Nov 26 2012
EXTENSIONS
a(16)-a(33) from Hiroaki Yamanouchi, Oct 03 2014
STATUS
approved