A063468 Number of Pythagorean triples in the range [1..n], i.e., the number of integer solutions to x^2 + y^2 = z^2 with 1 <= x,y,z <= n.

0, 0, 0, 0, 2, 2, 2, 2, 2, 4, 4, 4, 6, 6, 8, 8, 10, 10, 10, 12, 12, 12, 12, 12, 16, 18, 18, 18, 20, 22, 22, 22, 22, 24, 26, 26, 28, 28, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 36, 40, 42, 44, 46, 46, 48, 48, 48, 50, 50, 52, 54, 54, 54, 54, 62, 62, 62, 64, 64, 66, 66, 66, 68, 70

Offset

1,5

Links

Marius A. Burtea, Table of n, a(n) for n = 1..1000

Example

For n = 5 the Pythagorean triples are (3, 4, 5) and (4, 3, 5), so a (5) = 2.
For n = 10 the Pythagorean triples are (3, 4, 5), (4, 3, 5), (6, 8, 10) and (8, 6, 10), so a(10) = 4.
For n = 17 the Pythagorean triples are (3, 4, 5), (4, 5, 3), (5, 12, 13), (12, 5, 13), (6, 8, 10), (8, 6, 10), (8, 15, 17), (15, 8, 17), (9, 12, 15) and (12, 9, 15), so a(17) = 10.

Mathematica

nq[n_] := SquaresR[2, n^2]/4 - 1; Accumulate@ Array[nq, 80] (* Giovanni Resta, Jan 23 2020 *)

Prog

(Magma) [#[<x, y, Floor(Sqrt(x^2+y^2))>: x in [1..n], y in [1..n]| IsSquare(x^2+y^2) and Floor(Sqrt(x^2+y^2)) le n]:n in [1..74]]; // Marius A. Burtea, Jan 22 2020

Crossrefs

Cf. A062775, A211422.

a(n) = 2*partial sums of A046080(n).

Sequence in context: A260984 A108105 A321213 * A010336 A054537 A029104

Adjacent sequences: A063465 A063466 A063467 * A063469 A063470 A063471

Keyword

nonn

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 27 2001

Extensions

Corrected and extended by Vladeta Jovovic, Jul 28 2001

Status

approved