A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

12, 20, 24, 28, 36, 40, 45, 44, 48, 52, 60, 56, 60, 72, 72, 68, 75, 63, 84, 76, 80, 90, 84, 88, 105, 92, 105, 96, 120, 120, 104, 120, 108, 112, 132, 105, 116, 120, 144, 124, 144, 135, 132, 156, 136, 150, 126, 140, 168, 168, 180, 148, 175, 165, 152, 156, 168, 180

Offset

1,1

Comments

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.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pythagorean Triple

Formula

a(n) = sqrt(A386307(n)^2 - A386309(n)^2).

{A046084(n)} = {a(n)} union {A046087(n)} union {A386944(n)}.

Example

The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 12 is a term.

Maple

A386308:=proc(N) # To get all terms with hypotenuses <= N
    local i, l, m, u, v, r, x, y, z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1, floor(sqrt(N-u^2))) do
            x:=min(2*u*v, u^2-v^2);
            y:=max(2*u*v, u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z, y, x]};
            if gcd(u, v)=1 and is(u-v, odd) then
                l:=l union {seq([i*z, i*y, i*x], i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i, 2], i=1..nops(r));
end proc;
A386308(1000);

Crossrefs

Subsequence of A046084.

Cf. A366675, A380073, A386307, A386309, A386944.

Sequence in context: A332832 A065201 A136724 * A393184 A361868 A316597

Adjacent sequences: A386305 A386306 A386307 * A386309 A386310 A386311

Keyword

nonn

Author

Felix Huber, Aug 19 2025

Status

approved