Voozh

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..40.

Wikipedia, Positive Definite Matrix

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: