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; hence, this sequence of sides c is not increasing.
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 one-third of another one, for primitive integer-sided triangles.
All terms are cubes >= 8 (A000578). More generally, when c is the side of a primitive integer-sided triangles (a, b, c) whose angle B = m*C, then c = k^m, for some k >= 2.
Note that side c is never the largest 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, 3).
EXAMPLE
According to inequalities between a, b, c, there exist 3 types of such triangles:
a < c < b with the middle side c = 8 of the first triple (3, 10, 8).
c < a < b with the smallest side c = 27 of the 2nd triple (35, 48, 27).
c < b < a with the smallest side c = 216 of the 8th triple (539, 510, 216), the first of this type.
The smallest side c for which there exist two distinct triangles with B = 3*C is for a(4) = a(5) = 125, and these sides c belong respectively to triples (112, 195, 125) and (279, 280, 125).
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(q^3); end if;
end do;
end do;
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, May 07 2022
STATUS
approved