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

1,1

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.

Links

Table of n, a(n) for n=1..66.

APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Formula

a(n) = A343063(n, 2).

Example

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.

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)<sqrt(d) and sqrt(d)<a+c then print(sqrt(d)); end if;
end do;
end do;

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).

Sequence in context: A339858 A080714 A116368 * A290467 A007622 A180291

Adjacent sequences: A343062 A343063 A343064 * A343066 A343067 A343068

Keyword

nonn

Author

Bernard Schott, Apr 11 2021

Status

approved