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%I #19 Apr 12 2023 21:40:00
%S 10,48,132,195,280,357,504,510,665,792,840,840,1035,1288,1485,1575,
%T 1740,1848,1872,1890,2233,2496,2604,2610,2640,3003,3069,3520,3536,
%U 3885,4095,4368,4560,4620,4662,4680,5291,5712,5904,5928,6006,6579,6765,6992,7462,7480,7568,8037,8385,8415,8820
%N Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C.
%C 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.
%C In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
%C Equivalently, length of side opposite to the angle that is the triple of another one, for primitive integer-sided triangles.
%C Note that side b is never the smallest side of the triangle.
%C For the corresponding primitive triples and miscellaneous properties and references, see A353618.
%H The IMO Compendium, <a href="https://imomath.com/othercomp/Czs/CzsMO97.pdf">Problem 1</a>, 46th Czech and Slovak Mathematical Olympiad 1997.
%F a(n) = A353618(n, 2).
%e According to inequalities between a, b, c, there exist 3 types of such triangles:
%e a < c < b with the largest side b = 10 of the first triple (3, 10, 8).
%e c < a < b with the largest side b = 48 of the 2nd triple (35, 48, 27).
%e c < b < a with the middle side b = 510 of the 8th triple (539, 510, 216), the first of this type.
%e 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).
%p for b from 4 to 9000 do
%p for q from 2 to floor((b-1)^(1/3)) do
%p a := (b-q^3) * sqrt(1+b/q^3);
%p 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;
%p end do;
%p end do;
%Y Cf. A353618 (triples), A353619 (side a), this sequence (side b), A353621 (side c), A353622 (perimeter).
%Y Cf. A343065 (similar, but with B = 2*C).
%K nonn
%O 1,1
%A _Bernard Schott_, May 07 2022