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A353620
Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.
3
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
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
KEYWORD
nonn
AUTHOR
Bernard Schott, May 07 2022
STATUS
approved