A353622 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 3*C.

21, 110, 315, 432, 684, 720, 1100, 1265, 1560, 1612, 2106, 1885, 2240, 3255, 3040, 3808, 3485, 3978, 4760, 4403, 5040, 5130, 5719, 6669, 6232, 6440, 7560, 9030, 7896, 8008, 9504, 8855, 9752, 11891, 10557, 11270, 13200, 11800, 12825, 15300, 14575, 14040, 16016

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.

This sequence is nonincreasing: a(12) = 1885 < a(11) = 2106.

If in triangle ABC, B = 3*C, then the corresponding metric relation between sides is c*a^2 = (b-c)^2 * (b+c).

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

Links

Table of n, a(n) for n=1..43.

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

Formula

a(n) = A353618(n, 1) + A353618(n, 2) + A353618(n, 3).

a(n) = A353619(n) + A353620(n) + A353621(n).

Example

According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 21 with a < c < b for the first triple (3, 10, 8);
a(2) = 110 with c < a < b for the 2nd triple (35, 48, 27);
a(8) = 1265 with c < b < a for the 8th triple (539, 510, 216).

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(a+b+q^3);
    end if;
  end do;
end do;

Prog

(PARI) lista(nn) = {for (b = 4 , nn, for (q = 2 , sqrtnint(b-1, 3), if (issquare(z=1+b/q^3), a = (b-q^3) * sqrtint(numerator(z))/sqrtint(denominator(z)); if ((q^3 < b) && (gcd([a, b, q])==1) && (b-q^3 < a) && (a < b+q^3), print1(a+b+q^3, ", "); ); ); ); ); } \\ Michel Marcus, May 11 2022

Crossrefs

Cf. A353618 (triples), A353619 (side a), A353620 (side b), A353621 (side c).

Cf. A343067 (similar, but with B = 2*C).

Sequence in context: A071333 A222347 A039611 * A069307 A157886 A039456

Adjacent sequences: A353619 A353620 A353621 * A353623 A353624 A353625

Keyword

nonn

Author

Bernard Schott, May 08 2022

Status

approved