A336754 Perimeters in increasing order of integer-sided triangles whose sides a < b < c are in arithmetic progression.

9, 12, 15, 15, 18, 18, 21, 21, 21, 24, 24, 24, 27, 27, 27, 27, 30, 30, 30, 30, 33, 33, 33, 33, 33, 36, 36, 36, 36, 36, 39, 39, 39, 39, 39, 39, 42, 42, 42, 42, 42, 42, 45, 45, 45, 45, 45, 45, 45, 48, 48, 48, 48, 48, 48, 48, 51, 51, 51, 51, 51, 51, 51, 51

Offset: 1

Author

Bernard Schott, Aug 31 2020

Keywords

• nonn

Comments

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

As perimeter = 3 * middle side, these perimeters p are all multiple of 3, and each term p appears floor((p-3)/6) = A004526((p-3)/3) consecutively.

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

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

Examples

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

References

• V. Lespinard and R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-290 p. 121, André Desvigne.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004526.

Cf. A336750 (triples), A336751 (smallest side), A307136 (middle side), A336753 (largest side), this sequence (perimeter), A024164 (number of such triangles whose perimeter = n), A336755 (primitive triples).

Cf. A335897 (perimeters when angles A, B and C are in arithmetic progression).

Programs

Maple

for b from 3 to 30 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

A336754[n_] := 3*Ceiling[2*Sqrt[n+Round[Sqrt[n]]]]; Array[A336754, 100] (* or *)
Flatten[Array[ConstantArray[3*#, Floor[(#-1)/2]] &, 19, 3]] (* Paolo Xausa, Feb 29 2024 *)

Formula

a(n) = A336750(n, 1) + A336750(n, 2) + A336750(n, 3).

a(n) = 3 * A307136(n).