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

1,5

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

All terms are either 0 or powers of 2 (A000079), and record values occur at a(1) = 0, a(3) = 1, a(5) = 2, and then a(A002110(k+1)*2) = 2^k = A000079(k) for all k >= 2. - Charles L. Hohn, Jul 08 2026

Links

Frank M Jackson, Table of n, a(n) for n = 1..10000

Ron Knott, Pythagorean Triples and Online Calculators

Formula

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

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}] (* Frank M Jackson, Jun 12 2017 *)
(* Alternative: *)
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 *)

Prog

(PARI) a(n)={if(n<2 || n%4==2, return(0)); my(v=Vec(factor(n))[1]); 2^(#v-if(vecmax(v%4)==1, 0, 1))} \\ Charles L. Hohn, Jun 21 2026

Crossrefs

Cf. A024361, A024362, A000079, A002110.

Sequence in context: A213629 A357380 A278522 * A050600 A129691 A280750

Adjacent sequences: A024360 A024361 A024362 * A024364 A024365 A024366

Keyword

nonn,easy,changed

Author

David W. Wilson

Status

approved