A024164 Number of integer-sided triangles with sides a,b,c, a<b<c, a+b+c=n such that c - b = b - 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

1,15

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)

Links

Table of n, a(n) for n=1..102.

Wikipedia, Integer Triangle

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

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

Example

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).

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 *)

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).

Sequence in context: A033719 A171608 A307985 * A138805 A316400 A061897

Adjacent sequences: A024161 A024162 A024163 * A024165 A024166 A024167

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved