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A343065 Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

%I A343065 #14 Apr 22 2021 11:26:35
%S A343065 6,12,20,30,42,56,15,72,90,110,132,35,156,182,210,240,63,28,272,306,
%T A343065 342,40,380,99,420,462,506,552,143,600,70,650,702,756,45,195,88,812,
%U A343065 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 A343065 Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.
%C A343065 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 A343065 This sequence is not increasing because a(7) = 15 < a(6) = 56 (A106430).
%C A343065 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 A343065 Equivalently, length of side opposite to the greater of the two angles, one being the double of the other.
%C A343065 For the corresponding primitive triples and miscellaneous properties and references, see A343063.
%H A343065 APMEP, <a href="https://www.apmep.fr/IMG/pdf/Lyon_septembre_1937.pdf">Baccalauréat Mathématiques, Lyon, Septembre 1937</a>.
%F A343065 a(n) = A343063(n, 2).
%e A343065 According to inequalities between a, b, c, there exist 3 types of such triangles:
%e A343065 c < a < b for the first triple (5, 6, 4) with b = 6.
%e A343065 c < b < a for the second triple (16, 15, 9) with b = 15.
%e A343065 a < c < b for the seventh triple (7, 12, 9) with b = 12.
%p A343065 for a from 2 to 100 do
%p A343065 for c from 3 to floor(a^2/2) do
%p A343065 d := c*(a+c);
%p A343065 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 A343065 end do;
%p A343065 end do;
%Y A343065 Cf. A335895 (similar for A < B < C in arithmetic progression).
%Y A343065 Cf. A343063 (triples), A343064 (side a), A343066 (side c), A343067 (perimeter).
%Y A343065 Cf. A106420 (sides b sorted on perimeter), A106430 (sides b in increasing order).
%K A343065 nonn
%O A343065 1,1
%A A343065 _Bernard Schott_, Apr 11 2021