A335894 Smallest side of integer-sided primitive triangles whose angles A < B < C are in arithmetic order.

3, 5, 7, 8, 5, 16, 11, 24, 7, 33, 13, 35, 16, 39, 9, 56, 32, 45, 17, 63, 40, 51, 11, 85, 19, 80, 55, 57, 40, 77, 24, 95, 13, 120, 23, 120, 65, 88, 69, 91, 56, 115, 25, 143, 75, 112, 15, 161, 104, 105, 32, 175

Offset: 1

Author

Bernard Schott, Jul 15 2020

Keywords

• nonn

Comments

The triples of sides (a,b,c) 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 (see last example). This sequence lists the a's.

Equivalently, lengths of the smallest side a of primitive non-equilateral triangles that have an angle of Pi/3; indeed, this side is opposite to the smallest angle A.

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

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

When (a, b, c) is a triple with a < c/2, then (c-a, b, c) is the following triple because if b^2 = a^2 - a*c + c^2 then also b^2 = (c-a)^2 - (c-a)*c + c^2; hence, for each pair (b,c), there exist two distinct triangles whose smallest sides a_1 and a_2 satisfy a_1 + a_2 = c (see first example).

Examples

			For the pair b = 7, c = 8 the two corresponding values of a are 3 and 5 with 3 + 5 = 8 = c because:
    7^2 = 3^2 - 3*8 + 8^2, with triple (3, 7, 8),
    7^2 = 5^2 - 5*8 + 8^2, with triple (5, 7, 8).
For b = 91, there exist four corresponding values of a, two for b = 91 and c = 96 that are 11 and 85 with 11 + 85 = 96 = c, and two for b = 91 and c = 99 that are 19 and 80 with 19 + 80 = 99 = c; also these four smallest sides are ordered 11, 85, 19, 80 in the data because:
    91^2 = 11^2 - 11*96 + 96^2, with triple (11, 91, 96),
    91^2 = 85^2 - 85*96 + 96^2, with triple (85, 91, 96),
    91^2 = 19^2 - 19*99 + 99^2, with triple (19, 91, 99),
    91^2 = 80^2 - 80*99 + 99^2, with triple (80, 91, 99).
		

References

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

Crossrefs

Cf. A335893 (triples), A335895 (middle side), A335896 (largest side), A335897 (perimeter).

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,c-a); 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, ", ", c-a, ", "); ); ); ); ); } \\ Michel Marcus, Jul 16 2020

Formula

a(n) = A335893(n, 1).

a is such that a^2 - c*a + c^2 - b^2 = 0 with gcd(a,b) = 1 and a < b.