A387908 a(n) is the number of distinct integer-sided triangles with perimeter 6*n and integer area whose sides are in arithmetic progression.

0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 4, 0, 4, 0, 1, 1, 1, 0, 1, 1, 1, 1, 4, 1, 1, 0, 4, 1, 1, 0, 1, 0, 1, 2, 1, 0, 4, 0, 1, 0, 4, 1, 1, 0, 1, 1, 4, 1, 1, 1, 1, 1, 1, 0, 4, 0, 1, 1, 4, 0, 4, 1, 4, 1, 1, 0, 1, 0, 4, 0, 4, 0

Offset

0,15

Comments

Integer-sided triangles with perimeter p that is not divisible by 6 cannot have side lengths in arithmetic progression and integer area. Proof: Let the three sides be in arithmetic progression: (x, y, z) = (p/3 - d, p/3, p/3 + d), where p is the perimeter. By Heron's formula, the area A is A = sqrt((p/2)*(p/6 + d)*(p/6)*(p/6 - d)). If p is not divisible by 6, then the radicand is not a perfect square, so A is not an integer. Hence, no integer-sided triangle with sides in arithmetic progression can have integer area when 6 does not divide p.

There is no triangle with integer area for d = 0 because A = sqrt(3*n*n*n*n) = sqrt(3)*n^2.

Right triangles counted in this sequence are of the form (3*k, 4*k, 5*k), where k is a positive integer. In terms with even n >= 2 therefore one right triangle is counted.

Observation: For n <= 50000, a(n) takes the values {0, 1, 2, 4, 7, 13, 22, 37, 40, 67}.

Links

Felix Huber, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Arithmetic Progression

Eric Weisstein's World of Mathematics, Heronian Triangle

Formula

a(n) <= min(A024164(6*n), A051516(6*n)) for n >= 1.

a(2*k) >= 1 for positive integer k.

Example

The a(14) = 4 distinct integer-sided triangles with perimeter 14*6 = 84 and integer area whose sides are in arithmetic progression are (15, 28, 41) with area 126, (17, 28, 39) with area 210, (21, 28, 35) with area 294 and (26, 28, 30) with area 336.

Maple

A387908:=proc(n)
    local a, d;
    a:=0;
    for d to n-1 do
        if issqr((n^2-d^2)/3) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A387908(n), n=0..87);

Mathematica

A387908[n_]:=Module[{a = 0, d}, For[d = 0, d <= n - 1, d++, If[IntegerQ[Sqrt[(n^2 - d^2)/3]], a = a + 1]]; a]; Array[A387908, 88, 0] (* James C. McMahon, Oct 15 2025 *)

Crossrefs

Cf. A008588, A024164, A051516, A334277.

Sequence in context: A036861 A120324 A136630 * A275714 A111728 A359010

Adjacent sequences: A387905 A387906 A387907 * A387909 A387910 A387911

Keyword

nonn,easy

Author

Felix Huber, Sep 27 2025

Status

approved