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A024363 Number of primitive Pythagorean triangles with side n.

%I A024363 #24 Jun 19 2017 09:11:02
%S A024363 0,0,1,1,2,0,1,1,1,0,1,2,2,0,2,1,2,0,1,2,2,0,1,2,2,0,1,2,2,0,1,1,2,0,
%T A024363 2,2,2,0,2,2,2,0,1,2,2,0,1,2,1,0,2,2,2,0,2,2,2,0,1,4,2,0,2,1,4,0,1,2,
%U A024363 2,0,1,2,2,0,2,2,2,0,1,2,1,0,1,4,4,0,2,2,2,0,2,2,2,0,2,2,2,0,2
%N A024363 Number of primitive Pythagorean triangles with side n.
%C A024363 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times AUBUC takes value n.
%C A024363 Using Euclidean parameters (x, y) with x > y to generate primitive Pythagorean triples to capture all occurrences of side n, the Mma program below must allow the x parameter to iterate at least (n+1)/2 times. - _Frank M Jackson_, Jun 12 2017
%H A024363 Frank M Jackson, <a href="/A024363/b024363.txt">Table of n, a(n) for n = 1..10000</a>
%H A024363 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%F A024363 a(n)=0 for n=1 and n=2 (mod 4)=A016825. a(n)=A024361(n)+A024362(n). - _Lekraj Beedassy_, Dec 01 2003
%t A024363 lst={}; xmax=51; Do[If[GCD[x, y]==1&&OddQ[x+y], AppendTo[lst, Sort@{x^2-y^2, 2 x*y, x^2+y^2}]], {x, xmax}, {y, x}]; BinCounts[Select[Flatten@lst, #<2xmax &], {1, 2(xmax-1), 1}] (* or *)
%t A024363 a[n_] := Block[{x, y, s = List@ ToRules@ Reduce[(x^2-y^2 == n^2 || x^2 + y^2 == n^2) && x>y>0, {x, y}, Integers]}, If[s == {}, 0, Length@ Select[ {x, y} /. s, GCD @@ # == 1 &]]]; Array[a, 99] (* _Giovanni Resta_, Jun 19 2017 *)
%Y A024363 Cf. A024361, A024362.
%K A024363 nonn
%O A024363 1,5
%A A024363 _David W. Wilson_