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 the side c.
This sequence is nonincreasing: a(12) = 1885 < a(11) = 2106.
If in triangle ABC, B = 3*C, then the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
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
EXAMPLE
According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 21 with a < c < b for the first triple (3, 10, 8);
a(2) = 110 with c < a < b for the 2nd triple (35, 48, 27);
a(8) = 1265 with c < b < a for the 8th triple (539, 510, 216).
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(a+b+q^3);
end if;
end do;
end do;
PROG
(PARI) lista(nn) = {for (b = 4 , nn, for (q = 2 , sqrtnint(b-1, 3), if (issquare(z=1+b/q^3), a = (b-q^3) * sqrtint(numerator(z))/sqrtint(denominator(z)); if ((q^3 < b) && (gcd([a, b, q])==1) && (b-q^3 < a) && (a < b+q^3), print1(a+b+q^3, ", "); ); ); ); ); } \\ Michel Marcus, May 11 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, May 08 2022
STATUS
approved