A336757 Number of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression with a perimeter = n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 5, 0, 0, 2, 0, 0, 6, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 8, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 5, 0, 0, 11, 0, 0, 4

Offset: 1

Author

Bernard Schott, Sep 20 2020

Keywords

• nonn

Comments

Equivalently: number of primitive integer-sided triangles such that b = (a+c)/2 with a < c and perimeter = n.

As the perimeter of these triangles = 3*b where b is the middle side, a(n) >= 1 iff n = 3*b, with b >= 3.

When b is prime, all the triangles of perimeter n = 3*b are primitive, hence in this case: a(n) = A024164(n).

For the corresponding triples (primitive or not), miscellaneous properties and references, see A336750.

Examples

			a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for the Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
a(18) = 1 for the triple (5, 6, 7); the other triple (4, 6, 8) corresponding to a perimeter = 18 is not a primitive triple.
		

Crossrefs

Cf. A336750 (triples, primitive or not), A336755 (primitive triples), A336756 (perimeters of primitive triangles).

Cf. A024164 (number of such triangles, primitive or not).

Similar sequences: A005044 (integer-sided triangles), A024155 (right triangles), A070201 (with integral inradius).

Cf. A000010, A023022.

Formula

For n = 3*b, b >= 3, a(n) = A023022(b) = A000010(b)/2, otherwise a(n) = 0.