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

1,2

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:
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 *)

Crossrefs

Sequence in context: A051943 A059306 A192023 * A297507 A163655 A041190

Adjacent sequences: A219690 A219691 A219692 * A219694 A219695 A219696

Keyword

nonn

Author

W. Edwin Clark, Nov 25 2012

Status

approved