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A353620 Side b of primitive integer-sided triangles (a, b, c) whose angle B = 3*C. 3
10, 48, 132, 195, 280, 357, 504, 510, 665, 792, 840, 840, 1035, 1288, 1485, 1575, 1740, 1848, 1872, 1890, 2233, 2496, 2604, 2610, 2640, 3003, 3069, 3520, 3536, 3885, 4095, 4368, 4560, 4620, 4662, 4680, 5291, 5712, 5904, 5928, 6006, 6579, 6765, 6992, 7462, 7480, 7568, 8037, 8385, 8415, 8820 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The triples (a, b, c) are displayed in increasing order of side b, and if sides b coincide then in increasing order of the side c.

In the case B = 3*C, the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).

Equivalently, length of side opposite to the angle that is the triple of an other one, for primitive integer-sided triangle.

Note that side b is never the smallest side of the triangle.

For the corresponding primitive triples and miscellaneous properties and references, see A353618.

LINKS

Table of n, a(n) for n=1..51.

The IMO Compendium, Problem 1, 46th Czech and Slovak Mathematical Olympiad 1997.

FORMULA

a(n) = A353618(n, 2).

EXAMPLE

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

a < c < b with the largest side b = 10 of the first triple (3, 10, 8).

c < a < b with the largest side b = 48 of the 2nd triple (35, 48, 27).

c < b < a with the middle side b = 510 of the 8th triple (539, 510, 216), the first of this type.

The first side b for which there exist two distinct triangles with B = 3*C is for a(11) = a(12) = 840, and these sides b belong respectively to triples (923, 840, 343) and (533, 840, 512).

MAPLE

for b from 4 to 9000  do

  for q from 2 to floor((b-1)^(1/3)) do

a := (b-q^3) * sqrt(1+b/q^3);

if a= floor(a) and q^3 < b and igcd(a, b, q)=1 and (b-q^3) < a and a < b+q^3 then print(b); end if;

end do;

end do;

CROSSREFS

Cf. A353618 (triples), A353619 (side a), this sequence (side b), A353621 (side c), A353622 (perimeter).

Cf. A343065 (similar, but with B = 2*C).

Sequence in context: A121073 A210371 A195023 * A277229 A163724 A271638

Adjacent sequences:  A353617 A353618 A353619 * A353621 A353622 A353623

KEYWORD

nonn

AUTHOR

Bernard Schott, May 07 2022

STATUS

approved

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Last modified August 17 05:07 EDT 2022. Contains 356184 sequences. (Running on oeis4.)