A219693 The number of symmetric positive definite 2 X 2 matrices whose entries are integers of absolute value at most n.

1, 10, 31, 68, 133, 226, 351, 512, 721, 986, 1303, 1676, 2125, 2642, 3231, 3896, 4665, 5522, 6479, 7532, 8701, 9986, 11383, 12896, 14553, 16354, 18287, 20364, 22605, 24994, 27543, 30248, 33145, 36226, 39479, 42908, 46557, 50402, 54439, 58680

Offset: 1

Author

W. Edwin Clark, Nov 25 2012

Keywords

• nonn

Comments

A symmetric matrix [[a,c],[c,b]] is positive definite if and only if a > 0 and ab - c^2 > 0. So a(n) is also the number of triples (a,b,c) satisfying these inequalities with a,b,c having absolute value at most n.

Links

• Wikipedia, Positive Definite Matrix

Programs

Maple

  a:=proc(n)
local x,y,z,count;
count:=0;
for x from 1 to n do
for y from 1 to n do
for z from -n to n do
if x>0 and x*y > z^2 then count:=count+1; fi;
od:
od:
od:
count;
end:

Mathematica

Table[cnt = 0; Do[If[a*b > c^2, cnt++], {a, n}, {b, n}, {c, -n, n}]; cnt, {n, 40}] (* T. D. Noe, Nov 26 2012 *)