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

Author

Seiichi Manyama, Feb 15 2022

Keywords

• nonn
• tabf

Examples

			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),
  ...
		

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.

Crossrefs

Cf. A291694.

Programs

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

Extensions

More terms from Chai Wah Wu, Feb 16 2022