A335897 Perimeters of primitive integer-sided triangles whose angles A < B < C are in arithmetic order.

18, 20, 35, 36, 45, 56, 77, 90, 84, 110, 104, 126, 120, 143, 135, 182, 176, 189, 170, 216, 210, 221, 198, 272, 209, 270, 264, 266, 260, 297, 252, 323, 273, 380, 299, 396, 351, 374, 368, 390, 378, 437, 350, 468, 425, 462, 360, 506, 494, 495, 432, 575, 476, 585, 464, 630

Offset: 1

Author

Bernard Schott, Jul 17 2020

Keywords

• nonn

Comments

The triples of sides (a,b,c) of A335893 with a < b < c are in nondecreasing order of middle side, and if middle sides coincide, then by increasing order of the largest side, and when largest sides coincide, then by increasing order of the smallest side. This sequence lists the sums a+b+c (see last example).

Equivalently, perimeters of primitive non-equilateral triangles that have an angle of Pi/3.

Also, sum a+b+c of the solutions of the Diophantine equation b^2 = a^2 - b*c + c^2 with gcd(a,b) = 1 and a < b.

For the corresponding primitive triples and miscellaneous properties and references, see A335893.

This sequence is not increasing. For example, a(8) = 90 for triangle with middle side = 31 while a(9) = 84 for triangle with middle side = 37.

Examples

			For b = 7 and c = 8, the two corresponding triangles satisfy:
   7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8) and perimeter = 18,
   7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8) and perimeter = 20.
For b = 91, there exist four corresponding triangles, two for b = 91 and c = 96 and two for b = 91 and c = 99; the four corresponding perimeters are ordered 198, 272, 209, 270 in the data because:
   91^2 = 11^2 -11*96 +96^2, with triple (11, 91, 96) and perimeter 11+91+96 = 198,
   91^2 = 85^2 -85*96 +96^2, with triple (85, 91, 96) and perimeter 85+91+96 = 272,
   91^2 = 19^2 -19*99 +99^2, with triple (19, 91, 99) and perimeter 19+91+99 = 209,
   91^2 = 80^2 -80*99 +99^2, with triple (80, 91, 99) and perimeter 80+91+99 = 270.
		

References

• V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-298 p. 124, André Desvigne.

Crossrefs

Cf. A335893 (triples), A335894 (smallest side), A335895 (middle side), A335896 (largest side).

Programs

Maple

for b from 3 to 250 by 2 do
for c from b+1 to 6*b/5 do
a := (c - sqrt(4*b^2-3*c^2))/2;
if gcd(a,b)=1 and issqr(4*b^2-3*c^2) then print(a+b+c,2*c-a+b); end if;
end do;
end do;

PARI

lista(nn) = {forstep(b=1, nn, 2, for(c=b+1, 6*b\5, if (issquare(d=4*b^2 - 3*c^2), my(a = (c - sqrtint(d))/2); if ((denominator(a)==1) && (gcd(a, b) == 1), print1(a+b+c, ", ", 2*c-a+b, ", "); ); ); ); ); } \\ Michel Marcus, Jul 17 2020

Formula

a(n) = A335893(n, 1) + A335893(n, 2) + A335893(n, 3).

a(n) = A335894(n) + A335895(n) + A335896(n).