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A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), ordered by increasing hypotenuse (A386943).

%I A386944 #9 Aug 27 2025 22:35:14
%S A386944 8,16,24,30,32,36,48,48,42,60,70,64,80,72,96,96,90,84,108,120,100,112,
%T A386944 126,120,110,140,135,128,160,154,168,160,144,144,192,198,192,180,182,
%U A386944 216,224,168,216,240,196,200,234,224,252,189,240,210,286,288,220,280,280
%N A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), ordered by increasing hypotenuse (A386943).
%C A386944 In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
%H A386944 Felix Huber, <a href="/A386944/b386944.txt">Table of n, a(n) for n = 1..10000</a>
%H A386944 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A386944 a(n) = sqrt(A386943(n)^2 - A386945(n)^2).
%F A386944 {A046084(n)} = {a(n)} union {A046087(n)} union {A386308(n)}.
%e A386944 The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 8 is a term.
%p A386944 A386944:=proc(N) # To get all hypotenuses <= N
%p A386944     local i,l,u,v;
%p A386944     l:=[];
%p A386944     for u from 2 to floor(sqrt(N-1)) do
%p A386944         for v to min(u-1,floor(sqrt(N-u^2))) do
%p A386944             if gcd(u,v)>1 or is(u-v,even) then
%p A386944                 l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
%p A386944             fi
%p A386944         od
%p A386944     od;
%p A386944     l:=sort(l);
%p A386944     return seq(l[i,2],i=1..nops(l));
%p A386944 end proc;
%p A386944 A386944(296);
%Y A386944 Subsequence of A046084.
%Y A386944 Cf. A046087, A366675, A380073, A386308, A386943, A386945.
%K A386944 nonn,easy,new
%O A386944 1,1
%A A386944 _Felix Huber_, Aug 24 2025