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A219693 The number of symmetric positive definite 2 X 2 matrices whose entries are integers of absolute value at most n. 1
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:

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 A283846 A163655

Adjacent sequences:  A219690 A219691 A219692 * A219694 A219695 A219696

KEYWORD

nonn

AUTHOR

W. Edwin Clark, Nov 25 2012

STATUS

approved

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Last modified July 16 09:13 EDT 2020. Contains 335784 sequences. (Running on oeis4.)