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A343066 Side c of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

%I A343066 #8 Apr 28 2021 09:28:46
%S A343066 4,9,16,25,36,49,9,64,81,100,121,25,144,169,196,225,49,16,256,289,324,
%T A343066 25,361,81,400,441,484,529,121,576,49,625,676,729,25,169,64,784,841,
%U A343066 900,961,225,1024,1089,100,1156,1225,49,289,1296,121,1369,1444,1521,361,1600,1681,36,1764,169,1849,81,441,1936,2025,196
%N A343066 Side c of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.
%C A343066 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 A343066 This sequence is not increasing because a(7) = 9 < a(6) = 49.
%C A343066 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 A343066 All terms are perfect squares >= 4.
%C A343066 For the corresponding primitive triples and miscellaneous properties and references, see A343063.
%H A343066 APMEP, <a href="https://www.apmep.fr/IMG/pdf/Lyon_septembre_1937.pdf">Baccalauréat Mathématiques, Lyon, Septembre 1937</a>.
%F A343066 a(n) = A343063(n, 3)
%e A343066 According to inequalities between a, b, c, there exist 3 types of such triangles:
%e A343066 c = 4 with c < a < b for the first triple (5, 6, 4).
%e A343066 c = 9 with c < b < a for the seventh triple (16, 15, 9).
%e A343066 c = 16 with a < c < b for the third triple (9, 20, 16).
%p A343066 for a from 2 to 100 do
%p A343066 for c from 3 to floor(a^2/2) do
%p A343066 d := c*(a+c);
%p A343066 if issqr(d) and igcd(a,sqrt(d),c)=1 and abs(a-c)<sqrt(d) and sqrt(d)<a+c then print(c); end if;
%p A343066 end do;
%p A343066 end do;
%Y A343066 Cf. A335896 (similar for A < B < C in arithmetic progression).
%Y A343066 Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343067 (perimeter).
%K A343066 nonn
%O A343066 1,1
%A A343066 _Bernard Schott_, Apr 12 2021