%I #14 Apr 22 2021 11:26:35

%S 6,12,20,30,42,56,15,72,90,110,132,35,156,182,210,240,63,28,272,306,

%T 342,40,380,99,420,462,506,552,143,600,70,650,702,756,45,195,88,812,

%U 870,930,992,255,1056,1122,130,1190,1260,77,323,1332,154,1406,1482,1560,399,1640,1722,66,1806,208,1892,117,483,1980,2070,238

%N Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

%C The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.

%C This sequence is not increasing because a(7) = 15 < a(6) = 56 (A106430).

%C If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.

%C Equivalently, length of side opposite to the greater of the two angles, one being the double of the other.

%C For the corresponding primitive triples and miscellaneous properties and references, see A343063.

%H APMEP, <a href="https://www.apmep.fr/IMG/pdf/Lyon_septembre_1937.pdf">Baccalauréat Mathématiques, Lyon, Septembre 1937</a>.

%F a(n) = A343063(n, 2).

%e According to inequalities between a, b, c, there exist 3 types of such triangles:

%e c < a < b for the first triple (5, 6, 4) with b = 6.

%e c < b < a for the second triple (16, 15, 9) with b = 15.

%e a < c < b for the seventh triple (7, 12, 9) with b = 12.

%p for a from 2 to 100 do

%p for c from 3 to floor(a^2/2) do

%p d := c*(a+c);

%p if issqr(d) and igcd(a,sqrt(d),c)=1 and abs(a-c)<sqrt(d) and sqrt(d)<a+c then print(sqrt(d)); end if;

%p end do;

%p end do;

%Y Cf. A335895 (similar for A < B < C in arithmetic progression).

%Y Cf. A343063 (triples), A343064 (side a), A343066 (side c), A343067 (perimeter).

%Y Cf. A106420 (sides b sorted on perimeter), A106430 (sides b in increasing order).

%K nonn

%O 1,1

%A _Bernard Schott_, Apr 11 2021