

A336754


Perimeters in increasing order of integersided triangles whose sides a < b < c are in arithmetic progression.


6



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
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OFFSET

1,1


COMMENTS

Equivalently: perimeters of integersided 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((p3)/6) = A004526((p3)/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 & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B290 p. 121, André Desvigne.


LINKS

Table of n, a(n) for n=1..64.


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 bfloor((b1)/2) to b1 do
c := 2*b  a;
print(a+b+c);
end do;
end do;


CROSSREFS

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



