A351372 Array of triples (x,y,z) satisfy the Diophantine equation (x+y)^2 + (y+z)^2 + (z+x)^2 = 12*x*y*z, 1 <= x <= y <= z. (sorted by z).

1, 1, 1, 1, 1, 3, 1, 3, 13, 1, 13, 61, 3, 13, 217, 1, 61, 291, 1, 291, 1393, 3, 217, 3673, 13, 61, 4683, 1, 1393, 6673, 13, 217, 16693, 1, 6673, 31971, 3, 3673, 62221, 61, 291, 106153, 1, 31971, 153181, 13, 4683, 360517, 1, 153181, 733933, 3, 62221, 1054081

Offset

1,6

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10002

Yasuaki Gyoda, Positive integer solutions to (x+y)^2+(y+z)^2+(z+x)^2=12xyz, arXiv:2109.09639 [math.NT], 2021.

Example

The array of triples begins:
  ( 1,    1,     1),
  ( 1,    1,     3),
  ( 1,    3,    13),
  ( 1,   13,    61),
  ( 3,   13,   217),
  ( 1,   61,   291),
  ( 1,  291,  1393),
  ( 3,  217,  3673),
  (13,   61,  4683),
  ( 1, 1393,  6673),
  (13,  217, 16693),
  ...

Prog

(PARI) N=5000;
for(k=1, N, for(j=1, k, for(i=1, j, if(i*j>k, break); if((i+j)^2+(j+k)^2+(k+i)^2==12*i*j*k, print1(i, ", ", j, ", ", k, ", ")))));
(Python)
from math import isqrt
from itertools import count, islice
def A351372_gen(): # generator of terms
    for z in count(1):
        z2 = z**2
        for y in range(1, z+1):
            a = isqrt(d := 3*y**2*(12*z2 - 4*z - 1) - 3*z2*(4*y + 1) - 2*y*z)
            if a**2 == d:
                x, r = divmod(12*y*z - 2*y - 2*z - 2*a, 4)
                if y <= x <= z and r == 0:
                    yield from (y, x, z)
A351372_list = list(islice(A351372_gen(), 21)) # Chai Wah Wu, Feb 16 2022

Crossrefs

Cf. A291694.

Sequence in context: A001351 A216021 A327149 * A356411 A355793 A173424

Adjacent sequences: A351369 A351370 A351371 * A351373 A351374 A351375

Keyword

nonn,tabf

Author

Seiichi Manyama, Feb 15 2022

Extensions

More terms from Chai Wah Wu, Feb 16 2022

Status

approved