A014498 Varying radii of inscribed circles within primitive Pythagorean triples as a function of increasing values of hypotenuse.

1, 2, 3, 3, 6, 5, 4, 10, 5, 12, 7, 15, 14, 6, 15, 20, 9, 21, 7, 18, 28, 11, 8, 21, 30, 35, 22, 9, 36, 24, 35, 13, 42, 33, 45, 10, 26, 40, 44, 15, 39, 11, 30, 45, 55, 56, 30, 63, 52, 12, 33, 66, 17, 63, 65, 72, 34, 13, 77, 60, 55, 70, 78, 19, 51, 14, 88, 39, 60, 77, 38, 91, 68, 90

Offset

1,2

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Saltire Software, Pythagorean Triples

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Formula

Arrange all primitive Pythagorean triples a, b, c by value of hypotenuse c, then by long leg b; for n-th value of c, sequence gives radius of largest inscribed circle, (a+b-c)/2.

a(n) = (A046086(n) + A046087(n) - A020882(n))/2 = A087459(n)/2.

a(n) = sqrt(A118961(n)*A118962(n)/2). - Lekraj Beedassy, May 07 2006

Maple

with(NumberTheory):
f := proc(N)
    local k, l, m, x, y, z;
    l := [];
    for m from 2 while m^2 + 1 <= N do
        for k from 1 to m - 1 do
            if igcd(m, k) > 1 then next end if;
            if (m - k) mod 2 = 0 then next end if;
            z := m^2 + k^2;
            if z > N then break end if;
            x := m^2 - k^2;  y := 2*m*k; l := [op(l), [z, max(x, y), min(x, y), m, k]]
        end do
    end do;
    sort(l, (u, v) -> u[1] < v[1] or (u[1] = v[1] and u[2] < v[2]));
end proc:
A014498 := proc(N) # All terms such that A020882(n) <= N
    local a, i, k, l, m, r;
    l := f(N); a := [];
    for i to nops(l) do
        m := l[i][4];  k := l[i][5]; r := m*k - k^2; a := [op(a), r]
    end do;
    a
end proc:
A014498(500); # Felix Huber, Jul 07 2026

Crossrefs

For ordered values of (a+b-c)/2 see A020888.

Cf. A046086, A046087, A020882, A087459.

Sequence in context: A361168 A101447 A119322 * A186286 A391259 A023821

Adjacent sequences: A014495 A014496 A014497 * A014499 A014500 A014501

Keyword

nonn,changed

Author

Ralph Peterson (ralphp(AT)library.nrl.navy.mil)

Extensions

More terms from Asher Auel May 05 2000

Extended by Ray Chandler, Mar 09 2004

Status

approved