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The number of 2 X 2 matrices with no real eigenvalues and whose entries are integers of absolute value at most n.
3

%I #44 Nov 21 2016 04:23:39

%S 14,148,642,1832,4246,8420,15202,25296,39742,59668,86338,120840,

%T 165174,220356,288322,370816,470254,587940,726994,888728,1076422,

%U 1292404,1539442,1819440,2136734,2493700,2893586,3339544,3835782,4384036,4990466,5656752,6388158

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

%H Hiroaki Yamanouchi and Chai Wah Wu, <a href="/A207259/b207259.txt">Table of n, a(n) for n = 1..1000</a> (terms for n = 1..100 from Hiroaki Yamanouchi)

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

%p a:=proc(n)

%p local x,y,z,w,Eig,count;

%p count:=0;

%p for x from -n to n do

%p for y from -n to n do

%p for z from -n to n do

%p for w from -n to n do

%p Eig:=LinearAlgebra:-Eigenvalues(Matrix([[x,y],[z,w]]));

%p if Im(Eig[1]) <> 0 then count:=count+1; fi;

%p od:

%p od:

%p od:

%p od:

%p count;

%p end:

%Y Cf. A003024, A219736, A219744.

%K nonn,easy

%O 1,1

%A _W. Edwin Clark_, Nov 26 2012

%E a(16)-a(33) from _Hiroaki Yamanouchi_, Oct 03 2014