A024363 Number of primitive Pythagorean triangles with side n.

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, 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, 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

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times AUBUC takes value n.

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

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A024361, A024362.

Programs

Mathematica

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 *)
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 *)

Formula

a(n)=0 for n=1 and n=2 (mod 4)=A016825. a(n)=A024361(n)+A024362(n). - Lekraj Beedassy, Dec 01 2003