cp's OEIS Frontend

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

%I A219736 #21 Nov 21 2016 04:23:47
%S A219736 67,477,1759,4729,10395,20141,35423,58225,90579,134813,193503,269785,
%T A219736 366267,486925,635199,815105,1030371,1286221,1586447,1937033,2342379,
%U A219736 2808221,3340239,3945361,4628467,5396781,6257039,7216457,8281579,9461805,10762495,12193873
%N A219736 The number of 2 X 2 matrices with all eigenvalues real and whose entries are integers with absolute value at most n.
%H A219736 Hiroaki Yamanouchi and Chai Wah Wu, <a href="/A219736/b219736.txt">Table of n, a(n) for n = 1..1000</a> (terms for n = 1..100 from Hiroaki Yamanouchi)
%F A219736 a(n) = (2*n+1)^4 - A207259(n).
%p A219736 a:=proc(n)
%p A219736 local x,y,z,w,Eig,count;
%p A219736 count:=0;
%p A219736 for x from -n to n do
%p A219736 for y from -n to n do
%p A219736 for z from -n to n do
%p A219736 for w from -n to n do
%p A219736 Eig:=LinearAlgebra:-Eigenvalues(Matrix([[x,y],[z,w]]));
%p A219736 if Im(Eig[1]) = 0  then count:=count+1; fi;
%p A219736 od:
%p A219736 od:
%p A219736 od:
%p A219736 od:
%p A219736 count;
%p A219736 end:
%Y A219736 Cf. A003024, A207259.
%K A219736 nonn,easy
%O A219736 1,1
%A A219736 _W. Edwin Clark_, Nov 26 2012
%E A219736 a(16)-a(32) from _Hiroaki Yamanouchi_, Oct 03 2014