A336753 Largest side of integer-sided triangles whose sides a < b < c are in arithmetic progression.

4, 5, 7, 6, 8, 7, 10, 9, 8, 11, 10, 9, 13, 12, 11, 10, 14, 13, 12, 11, 16, 15, 14, 13, 12, 17, 16, 15, 14, 13, 19, 18, 17, 16, 15, 14, 20, 19, 18, 17, 16, 15, 22, 21, 20, 19, 18, 17, 16, 23, 22, 21, 20, 19, 18, 17, 25, 24, 23, 22, 21, 20, 19, 18, 26, 25, 24, 23, 22, 21, 20, 19

Offset: 1

Author

Bernard Schott, Aug 25 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 perimeters coincide, then by increasing order of the smallest side. This sequence lists the c's.

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

c >= 4 and each largest side c appears floor((c-1)/3) = A002264(c-1) times but not consecutively.

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

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

Examples

			c = 4 only for the smallest triangle (2, 3, 4).
c = 5 only for Pythagorean triple (3, 4, 5).
c = 6 only for triple (4, 5, 6).
c = 7 for the two triples (3, 5, 7) and (5, 6, 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), A336751 (smallest side), A307136 (middle side), this sequence (largest side), A336754 (perimeter).

Cf. A335896 (largest 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(c);
end do;
end do;

Mathematica

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

Formula

a(n) = A336750(n, 3).