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

1,1

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.

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

Formula

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

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

Example

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).

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 *)

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).

Sequence in context: A259313 A170951 A044859 * A336756 A114306 A009188

Adjacent sequences: A336751 A336752 A336753 * A336755 A336756 A336757

Keyword

nonn

Author

Bernard Schott, Aug 31 2020

Status

approved