A336750 Triples for integer-sided triangles whose sides a < b < c are in arithmetic progression.

2, 3, 4, 3, 4, 5, 3, 5, 7, 4, 5, 6, 4, 6, 8, 5, 6, 7, 4, 7, 10, 5, 7, 9, 6, 7, 8, 5, 8, 11, 6, 8, 10, 7, 8, 9, 5, 9, 13, 6, 9, 12, 7, 9, 11, 8, 9, 10, 6, 10, 14, 7, 10, 13, 8, 10, 12, 9, 10, 11, 6, 11, 16, 7, 11, 15, 8, 11, 14, 9, 11, 13, 10, 11, 12, 7, 12, 17, 8, 12, 16

Offset

1,1

Comments

The triples are displayed in increasing order of perimeter, and if perimeters coincide then by increasing order of the smallest side, hence, each triple (a, b, c) is in increasing order.

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

As the perimeter of these triangles = 3*b, the triples are also displayed in increasing order of middle side.

When a < b < c are in arithmetic progression with b - a = c - b = x, then 1 <= x <= floor((b-1)/2), hence, there exist for each side b >= 3, floor((b-1)/2) = A004526(b) triangles whose sides a < b < c are in arithmetic progression.

The only right integer-sided triangles such that a < b < c are in arithmetic progression correspond to the Pythagorean triples (3k, 4k, 5k) with k > 0.

There do not exist triangles whose sides a < b < c and angles A < B < C are both in arithmetic progression.

Three geometrical properties about these triangles, even if they are not integer-sided:

1) tan(A/2) * tan(C/2) = 1/3,

2) r = h_b/3, where r is the inradius and h_b the length of the altitude through B,

3) The line (IG) is parallel to side (AC), where I is the incenter and G is the centroid of the triangle.

References

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-288 p. 120, André Desvigne.

Links

Michel Marcus, Table of n, a(n) for n = 1..16650 (Rows 1 to 5550)

Wikipedia, Integer triangle.

Formula

T(n,1) = A336751(n); T(n,2) = A307136(n); T(n,3) = A336753(n).

A336754(n) = T(n,1) + T(n,2) + T(n,3).

Example

The smallest such triangle is (2, 3, 4).
The only triangle with perimeter = 12 corresponds to the Pythagorean triple: (3, 4, 5).
There exist two triangles with perimeter = 15 corresponding to triples (3, 5, 7) and (4, 5, 6).
There exist also two triangles with perimeter = 18 corresponding to triples (4, 6, 8) and (5, 6, 7).
The table begins:
  2, 3, 4;
  3, 4, 5;
  3, 5, 7;
  4, 5, 6;
  4, 6, 8;
  5, 6, 7;
  4, 7, 10;
  5, 7, 9;
  6, 7, 8;

Maple

for b from 3 to 20 do
for a from b-floor((b-1)/2) to b-1 do
c := 2*b - a;
print(a, b, c);
end do;
end do;

Mathematica

Block[{nn = 12, a, b, c}, Reap[Do[Do[Sow@ {a, b, 2 b - a}, {a, b - Floor[(b - 1)/2], b - 1}], {b, 3, nn}]][[-1, 1]] ] // Flatten (* Michael De Vlieger, Oct 15 2020 *)

Prog

(PARI) tabf(nn) = {for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); print(a, " ", b, " ", c); ); ); } \\ Michel Marcus, Sep 08 2020

Crossrefs

Cf. A336751 (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter), A024164 (number of triangles with perimeter = n), A336755 (primitive triples), A336756 (perimeter of primitive triangles), A336757 (number of primitive triangles with perimeter = n).

Cf. A004526 (number of triangles with middle side = b).

Cf. A103605 (similar, with Pythagorean triples).

Cf. A335893 (similar, with A, B, C in arithmetic progression).

Sequence in context: A081399 A221108 A205554 * A336755 A214613 A364801

Adjacent sequences: A336747 A336748 A336749 * A336751 A336752 A336753

Keyword

nonn,tabf

Author

Bernard Schott, Aug 03 2020

Status

approved