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A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.

%I A343064 #23 May 09 2022 08:34:36
%S A343064 5,7,9,11,13,15,16,17,19,21,23,24,25,27,29,31,32,33,33,35,37,39,39,40,
%T A343064 41,43,45,47,48,49,51,51,53,55,56,56,57,57,59,61,63,64,65,67,69,69,71,
%U A343064 72,72,73,75,75,77,79,80,81,83,85,85,87,87,88,88,89,91,93,93,95,95,96,97,99
%N A343064 Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.
%C A343064 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 A343064 In this case, the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
%C A343064 Equivalently, length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.
%C A343064 For the corresponding primitive triples and miscellaneous properties and references, see A343063.
%H A343064 APMEP, <a href="https://www.apmep.fr/IMG/pdf/Lyon_septembre_1937.pdf">Baccalauréat Mathématiques, Lyon, Septembre 1937</a>.
%F A343064 a(n) = A343063(n, 1).
%e A343064 According to inequalities between a, b, c, there exist 3 types of such triangles:
%e A343064    c < a < b for the smallest side a = 5 and triple (5, 6, 4).
%e A343064 The first side a for which there exist two distinct triangles with B = 2C is for a = 33 with the two other types of examples,
%e A343064    c < b < a with triple (33, 28, 16),
%e A343064    a < c < b with triple (33, 272, 256).
%p A343064 for a from 2 to 100 do
%p A343064 for c from 3 to floor(a^2/2) do
%p A343064 d := c*(a+c);
%p A343064 if issqr(d) and  igcd(a,sqrt(d),c)=1 and abs(a-c)<sqrt(d) and sqrt(d)<a+c then print(a); end if;
%p A343064 end do;
%p A343064 end do;
%Y A343064 Cf. A353619 (similar, but with B = 3*C).
%Y A343064 Cf. A343063 (triples), this sequence (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).
%Y A343064 Cf. A106505 (sides a without repetition), A106506 (sides a sorted on perimeter).
%K A343064 nonn
%O A343064 1,1
%A A343064 _Bernard Schott_, Apr 10 2021