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

1,1

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.

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

Formula

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

Example

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).

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 *)

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).

Sequence in context: A056792 A292127 A227861 * A294991 A300118 A256544

Adjacent sequences: A336748 A336749 A336750 * A336752 A336753 A336754

Keyword

nonn

Author

Bernard Schott, Aug 15 2020

Status

approved