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A336754 Perimeters in increasing order of integer-sided 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Equivalently: perimeters of integer-sided 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((p-3)/6) = A004526((p-3)/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 B-290 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 b-floor((b-1)/2) to b-1 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

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Last modified April 17 09:29 EDT 2021. Contains 343064 sequences. (Running on oeis4.)