A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

15, 28, 45, 66, 91, 120, 40, 153, 190, 231, 276, 84, 325, 378, 435, 496, 144, 77, 561, 630, 703, 104, 780, 220, 861, 946, 1035, 1128, 312, 1225, 170, 1326, 1431, 1540, 126, 420, 209, 1653, 1770, 1891, 2016, 544, 2145, 2278, 299, 2415, 2556, 198, 684, 2701, 350, 2850, 3003, 3160

Offset

1,1

Comments

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

This sequence is nonincreasing: a(7) = 40 < a(6) = 120.

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

As the metric relation is equivalent to a = m^2 - k^2, b = m*k, c = k^2, with gcd(m,k) = 1 and k < m < 2k, so all terms are of the form m^2 + m*k = m * (m+k) with gcd(m,k) = 1 and k < m < 2k. These perimeters are in increasing order in A106499.

For the corresponding primitive triples and miscellaneous properties and references, see A343063.

Links

Table of n, a(n) for n=1..54.

Formula

a(n) = A343063(n, 1) + A343063(n, 2) + A343063(n, 3).

a(n) = A343064(n) + A343065(n) + A343066(n).

Example

According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 15 with c < a < b for the first triple (5, 6, 4);
a(7) = 40 with c < b < a for the seventh triple (16, 15, 9);
a(8) = 153 with a < c < b for the eighth triple (17, 72, 64).

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(a+sqrt(d)+c); end if;
end do;
end do;

Prog

(PARI) lista(nn) = {for (a = 2, nn, for (c = 3, a^2\2, my(d = c*(a+c)); if (issquare(d) && (gcd([a, sqrtint(d), c])==1) && (abs(a-c)<sqrtint(d)) && (sqrtint(d)<a+c),  print1(a+sqrtint(d)+c ", ")); ); ); } \\ Michel Marcus, May 12 2022

Crossrefs

Cf. A335897 (similar for A < B < C in arithmetic progression).

Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343066 (side c), A106499 (perimeters in increasing order).

Sequence in context: A043888 A350045 A357278 * A223442 A121594 A163286

Adjacent sequences: A343064 A343065 A343066 * A343068 A343069 A343070

Keyword

nonn

Author

Bernard Schott, Apr 15 2021

Status

approved