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

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 33, 35, 37, 39, 39, 40, 41, 43, 45, 47, 48, 49, 51, 51, 53, 55, 56, 56, 57, 57, 59, 61, 63, 64, 65, 67, 69, 69, 71, 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

Offset: 1

Author

Bernard Schott, Apr 10 2021

Keywords

• nonn

Comments

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.

In this case, the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.

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

For the corresponding primitive triples and miscellaneous properties and references, see A343063.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
   c < a < b for the smallest side a = 5 and triple (5, 6, 4).
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,
   c < b < a with triple (33, 28, 16),
   a < c < b with triple (33, 272, 256).
		

Links

• APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Crossrefs

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

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

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

Programs

Maple

for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and  igcd(a,sqrt(d),c)=1 and abs(a-c)
				

Formula

a(n) = A343063(n, 1).