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

4, 9, 16, 25, 36, 49, 9, 64, 81, 100, 121, 25, 144, 169, 196, 225, 49, 16, 256, 289, 324, 25, 361, 81, 400, 441, 484, 529, 121, 576, 49, 625, 676, 729, 25, 169, 64, 784, 841, 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

Offset: 1

Author

Bernard Schott, Apr 12 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.

This sequence is not increasing because a(7) = 9 < a(6) = 49.

If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.

All terms are perfect squares >= 4.

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 = 4 with c < a < b for the first triple (5, 6, 4).
c = 9 with c < b < a for the seventh triple (16, 15, 9).
c = 16 with a < c < b for the third triple (9, 20, 16).
		

Links

• APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Crossrefs

Cf. A335896 (similar for A < B < C in arithmetic progression).

Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343067 (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, 3)