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%I #15 May 13 2022 14:04:31
%S 21,110,315,432,684,720,1100,1265,1560,1612,2106,1885,2240,3255,3040,
%T 3808,3485,3978,4760,4403,5040,5130,5719,6669,6232,6440,7560,9030,
%U 7896,8008,9504,8855,9752,11891,10557,11270,13200,11800,12825,15300,14575,14040,16016
%N Perimeter of integer-sided primitive 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 the side c.
%C This sequence is nonincreasing: a(12) = 1885 < a(11) = 2106.
%C If in triangle ABC, B = 3*C, then the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).
%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, 1) + A353618(n, 2) + A353618(n, 3).
%F a(n) = A353619(n) + A353620(n) + A353621(n).
%e According to inequalities between a, b, c, there exist 3 types of such triangles:
%e a(1) = 21 with a < c < b for the first triple (3, 10, 8);
%e a(2) = 110 with c < a < b for the 2nd triple (35, 48, 27);
%e a(8) = 1265 with c < b < a for the 8th triple (539, 510, 216).
%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
%p print(a+b+q^3);
%p end if;
%p end do;
%p end do;
%o (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
%Y Cf. A353618 (triples), A353619 (side a), A353620 (side b), A353621 (side c).
%Y Cf. A343067 (similar, but with B = 2*C).
%K nonn
%O 1,1
%A _Bernard Schott_, May 08 2022
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