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A343063 Primitive triples (a, b, c) for integer-sided triangles whose angle B = 2*C.

%I A343063 #39 Apr 29 2021 04:36:59
%S A343063 5,6,4,7,12,9,9,20,16,11,30,25,13,42,36,15,56,49,16,15,9,17,72,64,19,
%T A343063 90,81,21,110,100,23,132,121,24,35,25,25,156,144,27,182,169,29,210,
%U A343063 196,31,240,225,32,63,49,33,28,16,33,272,256,35,306,289,37,342,324,39,40,25,39,380,361,40,99,81,41,420,400,43,462,441
%N A343063 Primitive triples (a, b, c) for integer-sided triangles whose angle B = 2*C.
%C A343063 This sequence is inspired by the problem of French Baccalauréat Mathématiques at Lyon in 1937 (see link).
%C A343063 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 A343063 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 A343063 This metric relation is equivalent to a = m^2 - k^2, b = m * k, c = k^2, gcd(m,k) = 1 and k < m < 2k; hence every c is a square number.
%C A343063 When A <> 45° and A <> 72°, table below shows there exist these 3 possible inequalities: c < b < a, c < a < b, a < c < b.
%C A343063    ------------------------------------------------------------------------
%C A343063    | A | 180 |   decr.  |   72    |   decr.   |   45    |   decr.   |  0  |
%C A343063    ------------------------------------------------------------------------
%C A343063    | B |  0  |   incr.  |   72    |   incr.   |   90    |   incr.   | 120 |
%C A343063    ------------------------------------------------------------------------
%C A343063    | C |  0  |   incr.  |   36    |   incr.   |   45    |   incr.   |  60 |
%C A343063    ------------------------------------------------------------------------
%C A343063    | < | No | c < b < a | c < b=a | c < a < b | c=a < b | a < c < b |  No |
%C A343063    ------------------------------------------------------------------------
%C A343063 where 'No' means there is no such corresponding triangle.
%C A343063 If (A,B,C) = (72,72,36) then a = b = c * (1+sqrt(5))/2 and isosceles ABC is not an integer-sided triangle.
%C A343063 If (A,B,C) = (45,90,45) then ABC is isosceles rectangle in B, so a = c with b = a*sqrt(2) and ABC is not an integer-sided triangle.
%D A343063 V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-336 p. 178, André Desvigne.
%H A343063 APMEP, <a href="https://www.apmep.fr/IMG/pdf/Lyon_septembre_1937.pdf">Baccalauréat Mathématiques, Lyon, Septembre 1937</a>.
%e A343063 The smallest such triangle is (5, 6, 4), of type c < a < b with 4*(5+4) = 6^2.
%e A343063 The 2nd triple is (7, 12, 9) of type a < c < b with 9*(7+9) = 16^2.
%e A343063 The 7th triple (16, 15, 9) is the first of type c < b < a with 9*(16+9) = 15^2.
%e A343063 The table begins:
%e A343063    5,  6,  4;
%e A343063    7, 12,  9;
%e A343063    9, 20, 16;
%e A343063   11, 30, 25,
%e A343063   13, 42, 36;
%e A343063   15, 56, 49;
%e A343063   16, 15,  9;
%e A343063   17, 72, 64;
%e A343063   ...
%p A343063 for a from 2 to 60 do
%p A343063 for c from 3 to floor(a^2/2) do
%p A343063 d := c*(a+c);
%p A343063 if igcd(a,sqrt(d),c)=1 and issqr(d) and abs(a-c)<sqrt(d) and sqrt(d)<a+c then print(a,sqrt(d),c); end if;
%p A343063 end do;
%p A343063 end do;
%Y A343063 Cf. A335893 (similar for A < B < C in arithmetic progression).
%Y A343063 Cf. A343064 (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).
%K A343063 nonn,tabf
%O A343063 1,1
%A A343063 _Bernard Schott_, Apr 04 2021