A343063 Primitive triples (a, b, c) for integer-sided triangles whose angle B = 2*C.

5, 6, 4, 7, 12, 9, 9, 20, 16, 11, 30, 25, 13, 42, 36, 15, 56, 49, 16, 15, 9, 17, 72, 64, 19, 90, 81, 21, 110, 100, 23, 132, 121, 24, 35, 25, 25, 156, 144, 27, 182, 169, 29, 210, 196, 31, 240, 225, 32, 63, 49, 33, 28, 16, 33, 272, 256, 35, 306, 289, 37, 342, 324, 39, 40, 25, 39, 380, 361, 40, 99, 81, 41, 420, 400, 43, 462, 441

Offset

1,1

Comments

This sequence is inspired by the problem of French Baccalauréat Mathématiques at Lyon in 1937 (see link).

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.

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

This metric relation is equivalent to a = m^2 - k^2, b = m * k, c = k^2, gcd(m,k) = 1 and k < m < 2k; hence every c is a square number.

When A <> 45° and A <> 72°, table below shows there exist these 3 possible inequalities: c < b < a, c < a < b, a < c < b.

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| A | 180 | decr. | 72 | decr. | 45 | decr. | 0 |

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| B | 0 | incr. | 72 | incr. | 90 | incr. | 120 |

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| C | 0 | incr. | 36 | incr. | 45 | incr. | 60 |

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| < | No | c < b < a | c < b=a | c < a < b | c=a < b | a < c < b | No |

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where 'No' means there is no such corresponding triangle.

If (A,B,C) = (72,72,36) then a = b = c * (1+sqrt(5))/2 and isosceles ABC is not an integer-sided triangle.

If (A,B,C) = (45,90,45) then ABC is isosceles rectangle in B, so a = c with b = a*sqrt(2) and ABC is not an integer-sided triangle.

References

V. Lespinard & R. Pernet, Trigonométrie, Classe de Mathématiques élémentaires, programme 1962, problème B-336 p. 178, André Desvigne.

Links

Table of n, a(n) for n=1..78.

APMEP, Baccalauréat Mathématiques, Lyon, Septembre 1937.

Example

The smallest such triangle is (5, 6, 4), of type c < a < b with 4*(5+4) = 6^2.
The 2nd triple is (7, 12, 9) of type a < c < b with 9*(7+9) = 16^2.
The 7th triple (16, 15, 9) is the first of type c < b < a with 9*(16+9) = 15^2.
The table begins:
   5,  6,  4;
   7, 12,  9;
   9, 20, 16;
  11, 30, 25,
  13, 42, 36;
  15, 56, 49;
  16, 15,  9;
  17, 72, 64;
  ...

Maple

for a from 2 to 60 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if igcd(a, sqrt(d), c)=1 and issqr(d) and abs(a-c)<sqrt(d) and sqrt(d)<a+c then print(a, sqrt(d), c); end if;
end do;
end do;

Crossrefs

Cf. A335893 (similar for A < B < C in arithmetic progression).

Cf. A343064 (side a), A343065 (side b), A343066 (side c), A343067 (perimeter).

Sequence in context: A201678 A245870 A351217 * A132324 A021643 A021181

Adjacent sequences: A343060 A343061 A343062 * A343064 A343065 A343066

Keyword

nonn,tabf

Author

Bernard Schott, Apr 04 2021

Status

approved