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

1,1

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.

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, 3).

Example

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).

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 *)

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).

Sequence in context: A016720 A104140 A003563 * A023833 A151551 A021690

Adjacent sequences: A336750 A336751 A336752 * A336754 A336755 A336756

Keyword

nonn

Author

Bernard Schott, Aug 25 2020

Status

approved