A343065 Side b of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

6, 12, 20, 30, 42, 56, 15, 72, 90, 110, 132, 35, 156, 182, 210, 240, 63, 28, 272, 306, 342, 40, 380, 99, 420, 462, 506, 552, 143, 600, 70, 650, 702, 756, 45, 195, 88, 812, 870, 930, 992, 255, 1056, 1122, 130, 1190, 1260, 77, 323, 1332, 154, 1406, 1482, 1560, 399, 1640, 1722, 66, 1806, 208, 1892, 117, 483, 1980, 2070, 238

Offset: 1

Author

Bernard Schott, Apr 11 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) = 15 < a(6) = 56 (A106430).

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

Equivalently, length of side opposite to the greater of the two angles, one being the double of the other.

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 first triple (5, 6, 4) with b = 6.
c < b < a for the second triple (16, 15, 9) with b = 15.
a < c < b for the seventh triple (7, 12, 9) with b = 12.
		

Links

• APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Crossrefs

Cf. A335895 (similar for A < B < C in arithmetic progression).

Cf. A343063 (triples), A343064 (side a), A343066 (side c), A343067 (perimeter).

Cf. A106420 (sides b sorted on perimeter), A106430 (sides b in increasing order).

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, 2).