A336751 Smallest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.

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

Offset: 1

Author

Bernard Schott, Aug 15 2020

Keywords

• nonn

Comments

The triples of sides (a,b,c) with a < b < c are in increasing order of perimeter = 3*b, and if perimeter coincide, then by increasing order of the smallest side. This sequence lists the a's.

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

a >= 2 and each side a appears a-1 times but not consecutively.

For each a = 3*k, k>=1, there exists exactly one right triangle (3*k, 4*k, 5*k) whose sides a < b < c are in arithmetic progression.

This sequence is not increasing a(6) = 5 for triangle with perimeter = 18 and a(7) = 4 for triangle with perimeter = 21. The smallest side is not an increasing function of the perimeter of these triangles.

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

Examples

			a = 2 for only the smallest triangle (2, 3, 4).
a = 3 for Pythagorean triple (3, 4, 5) and also for the second triangle (3, 5, 7).
		

References

• V. Lespinard & 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. A336750 (triples), this sequence (smallest side), A307136 (middle side), A336753 (largest side), A336754 (perimeter).

Cf. A335894 (smallest side when triangles angles 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);
end do;
end do;

Mathematica

Flatten[Array[Range[#-Floor[(#-1)/2], #-1] &, 20, 3]] (* Paolo Xausa, Feb 28 2024 *)

Formula

a(n) = A336750(n, 1).