A024164 Number of integer-sided triangles with sides a,b,c, a

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 5, 0, 0, 5, 0, 0, 6, 0, 0, 6, 0, 0, 7, 0, 0, 7, 0, 0, 8, 0, 0, 8, 0, 0, 9, 0, 0, 9, 0, 0, 10, 0, 0, 10, 0, 0, 11, 0, 0, 11, 0, 0, 12, 0, 0, 12, 0, 0, 13, 0, 0, 13, 0, 0, 14, 0, 0, 14, 0, 0, 15, 0, 0, 15, 0, 0, 16, 0, 0, 16

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

From Bernard Schott, Oct 10 2020: (Start)

Equivalently: number of integer-sided triangles whose sides a < b < c are in arithmetic progression with perimeter n.

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

All the perimeters are multiple of 3 because each perimeter = 3 * middle side b.

For each perimeter n = 12*k with k>0, there exists one and only one such right integer triangle whose triple is (3k, 4k, 5k).

For the corresponding primitive triples and miscellaneous properties and references, see A336750. (End)

Examples

			a(9) = 1 for the smallest such triangle (2, 3, 4).
a(12) = 1 for Pythagorean triple (3, 4, 5).
a(15) = 2 for the two triples (3, 5, 7) and (4, 5, 6).
		

Links

• Wikipedia, Integer Triangle
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Crossrefs

Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), this sequence (number of triangles whose perimeter = n), A336755 (primitive triples), A336756 (primitive perimeters), A336757 (number of primitive triangles with perimeter = n).

Cf. A005044 (number of integer-sided triangles with perimeter = n).

Programs

Mathematica

A024164[n_] := If[Mod[n, 3] == 0, Floor[(n - 3)/6], 0]; Array[A024164, 100] (* Wesley Ivan Hurt, Nov 01 2020 *)
LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{0,0,0,0,0,0,0,0,1},120] (* Harvey P. Dale, Jun 03 2021 *)

Formula

If n = 3*k, then a(n) = floor((n-3)/6) = A004526((n-3)/3), otherwise, a(3k+1) = a(3k+2) = 0. - Bernard Schott, Oct 10 2020

From Wesley Ivan Hurt, Nov 01 2020: (Start)

G.f.: x^9/((x^3 - 1)^2*(x^3 + 1)).

a(n) = a(n-3) + a(n-6) - a(n-9).

a(n) = (1 - ceiling(n/3) + floor(n/3)) * floor((n-3)/6). (End)

E.g.f.: (18 + (x - 6)*cosh(x) + (x - 3)*sinh(x) - exp(-x/2)*((9 + 3*exp(x) + x)*cos(sqrt(3)*x/2) + sqrt(3)*x*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, Feb 29 2024