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A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

%I A379830 #6 Jan 17 2025 19:38:51
%S A379830 0,0,1,0,1,0,1,0,2,2,1,0,1,0,1,0,2,0,4,0,1,0,1,0,2,3,1,2,1,0,1,0,5,0,
%T A379830 1,0,4,0,1,0,2,0,1,0,1,2,1,0,2,4,7,0,1,0,4,0,2,0,1,0,1,0,1,2,5,0,1,0,
%U A379830 1,0,1,0,8,0,1,3,1,0,1,0,2,6,1,0,1,0,1,0
%N A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.
%C A379830 The difference between the hypotenuse and the short leg of a primitive Pythagorean triple (p^2 - q^2, 2*p*q, p^2 + q^2) (where p > q are coprimes and not both odd) is d = max(2*q^2, (p - q)^2). For every of these primitive Pythagorean triples whose d divides n, there is a Pythagorean triple with w - u = n. Therefore d <= n and it follows that 1 <= q <= sqrt(n/2) and q + 1 <= p <= q + sqrt(n), which means that there is a finite number of Pythagorean triples with w - u = n.
%H A379830 Felix Huber, <a href="/A379830/b379830.txt">Table of n, a(n) for n = 0..10000</a>
%H A379830 Felix Huber, <a href="/A379830/a379830.txt">Pythagorean triples for which w - u = n</a>
%H A379830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%e A379830 The a(18) = 4 Pythagorean triples are (27, 36, 45), (16, 30, 34), (40, 42, 58), (7, 24, 25) because 45 - 27 = 34 - 16 = 58 - 40 = 25 - 7 = 18.
%e A379830 See also linked Maple program "Pythagorean triples for which w - u = n".
%p A379830 A379830:=proc(n)
%p A379830     local a,p,q;
%p A379830     a:=0;
%p A379830     for q to isqrt(floor(n/2)) do
%p A379830         for p from q+1 to q+isqrt(n) do
%p A379830             if igcd(p,q)=1 and (is(p,even) or is(q,even)) and n mod max((p-q)^2,2*q^2)=0 then
%p A379830                 a:=a+1
%p A379830             fi
%p A379830         od
%p A379830     od;
%p A379830     return a
%p A379830 end proc;
%p A379830 seq(A379830(n),n=0..87);
%Y A379830 Cf. A024361, A024362, A024363, A046079, A046080, A046081, A096033, A224921.
%K A379830 nonn
%O A379830 0,9
%A A379830 _Felix Huber_, Jan 07 2025