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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).
2
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
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
KEYWORD
nonn,tabf
AUTHOR
Seiichi Manyama, Feb 15 2022
EXTENSIONS
More terms from Chai Wah Wu, Feb 16 2022
STATUS
approved