login
Side a of primitive integer-sided triangles (a, b, c) whose angle B = 2*C.
6

%I #23 May 09 2022 08:34:36

%S 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 41,43,45,47,48,49,51,51,53,55,56,56,57,57,59,61,63,64,65,67,69,69,71,

%U 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 Side a of primitive integer-sided 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 In this case, the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.

%C Equivalently, length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

%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, 1).

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

%e c < a < b for the smallest side a = 5 and triple (5, 6, 4).

%e 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 c < b < a with triple (33, 28, 16),

%e a < c < b with triple (33, 272, 256).

%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(a); end if;

%p end do;

%p end do;

%Y Cf. A353619 (similar, but with B = 3*C).

%Y Cf. A343063 (triples), this sequence (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).

%Y Cf. A106505 (sides a without repetition), A106506 (sides a sorted on perimeter).

%K nonn

%O 1,1

%A _Bernard Schott_, Apr 10 2021