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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.

%I A063468 #17 Sep 08 2022 08:45:03
%S A063468 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,
%T A063468 20,22,22,22,22,24,26,26,28,28,30,32,34,34,34,34,36,36,36,36,36,40,42,
%U A063468 44,46,46,48,48,48,50,50,52,54,54,54,54,62,62,62,64,64,66,66,66,68,70
%N 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.
%H A063468 Marius A. Burtea, <a href="/A063468/b063468.txt">Table of n, a(n) for n = 1..1000</a>
%e A063468 For n = 5 the Pythagorean triples are (3, 4, 5) and (4, 3, 5), so a (5) = 2.
%e A063468 For n = 10 the Pythagorean triples are (3, 4, 5), (4, 3, 5), (6, 8, 10) and (8, 6, 10), so a(10) = 4.
%e A063468 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.
%t A063468 nq[n_] := SquaresR[2, n^2]/4 - 1; Accumulate@ Array[nq, 80] (* _Giovanni Resta_, Jan 23 2020 *)
%o A063468 (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
%Y A063468 Cf. A062775, A211422.
%Y A063468 a(n) = 2*partial sums of A046080(n).
%K A063468 nonn
%O A063468 1,5
%A A063468 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 27 2001
%E A063468 Corrected and extended by _Vladeta Jovovic_, Jul 28 2001