A353620 Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.

10, 48, 132, 195, 280, 357, 504, 510, 665, 792, 840, 840, 1035, 1288, 1485, 1575, 1740, 1848, 1872, 1890, 2233, 2496, 2604, 2610, 2640, 3003, 3069, 3520, 3536, 3885, 4095, 4368, 4560, 4620, 4662, 4680, 5291, 5712, 5904, 5928, 6006, 6579, 6765, 6992, 7462, 7480, 7568, 8037, 8385, 8415, 8820

Offset

1,1

Comments

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of side c.

In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).

Equivalently, length of side opposite to the angle that is the triple of another one, for primitive integer-sided triangles.

Note that side b is never the smallest side of the triangle.

For the corresponding primitive triples and miscellaneous properties and references, see A353618.

Links

Table of n, a(n) for n=1..51.

The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

Formula

a(n) = A353618(n, 2).

Example

According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the largest side b = 10 of the first triple (3, 10, 8).
c < a < b with the largest side b = 48 of the 2nd triple (35, 48, 27).
c < b < a with the middle side b = 510 of the 8th triple (539, 510, 216), the first of this type.
The first side b for which there exist two distinct triangles with B = 3*C is for a(11) = a(12) = 840, and these sides b belong respectively to triples (923, 840, 343) and (533, 840, 512).

Maple

for b from 4 to 9000  do
  for q from 2 to floor((b-1)^(1/3)) do
a := (b-q^3) * sqrt(1+b/q^3);
if a= floor(a) and q^3 < b and igcd(a, b, q)=1 and (b-q^3) < a and a < b+q^3 then print(b); end if;
end do;
end do;

Crossrefs

Cf. A353618 (triples), A353619 (side a), this sequence (side b), A353621 (side c), A353622 (perimeter).

Cf. A343065 (similar, but with B = 2*C).

Sequence in context: A121073 A210371 A195023 * A277229 A163724 A271638

Adjacent sequences: A353617 A353618 A353619 * A353621 A353622 A353623

Keyword

nonn

Author

Bernard Schott, May 07 2022

Status

approved