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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).

%I A351372 #37 Feb 16 2022 12:48:20
%S A351372 1,1,1,1,1,3,1,3,13,1,13,61,3,13,217,1,61,291,1,291,1393,3,217,3673,
%T A351372 13,61,4683,1,1393,6673,13,217,16693,1,6673,31971,3,3673,62221,61,291,
%U A351372 106153,1,31971,153181,13,4683,360517,1,153181,733933,3,62221,1054081
%N 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).
%H A351372 Chai Wah Wu, <a href="/A351372/b351372.txt">Table of n, a(n) for n = 1..10002</a>
%H A351372 Yasuaki Gyoda, <a href="https://arxiv.org/abs/2109.09639">Positive integer solutions to (x+y)^2+(y+z)^2+(z+x)^2=12xyz</a>, arXiv:2109.09639 [math.NT], 2021.
%e A351372 The array of triples begins:
%e A351372   ( 1,    1,     1),
%e A351372   ( 1,    1,     3),
%e A351372   ( 1,    3,    13),
%e A351372   ( 1,   13,    61),
%e A351372   ( 3,   13,   217),
%e A351372   ( 1,   61,   291),
%e A351372   ( 1,  291,  1393),
%e A351372   ( 3,  217,  3673),
%e A351372   (13,   61,  4683),
%e A351372   ( 1, 1393,  6673),
%e A351372   (13,  217, 16693),
%e A351372   ...
%o A351372 (PARI) N=5000;
%o A351372 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, ", ")))));
%o A351372 (Python)
%o A351372 from math import isqrt
%o A351372 from itertools import count, islice
%o A351372 def A351372_gen(): # generator of terms
%o A351372     for z in count(1):
%o A351372         z2 = z**2
%o A351372         for y in range(1,z+1):
%o A351372             a = isqrt(d := 3*y**2*(12*z2 - 4*z - 1) - 3*z2*(4*y + 1) - 2*y*z)
%o A351372             if a**2 == d:
%o A351372                 x, r = divmod(12*y*z - 2*y - 2*z - 2*a,4)
%o A351372                 if y <= x <= z and r == 0:
%o A351372                     yield from (y,x,z)
%o A351372 A351372_list = list(islice(A351372_gen(),21)) # _Chai Wah Wu_, Feb 16 2022
%Y A351372 Cf. A291694.
%K A351372 nonn,tabf
%O A351372 1,6
%A A351372 _Seiichi Manyama_, Feb 15 2022
%E A351372 More terms from _Chai Wah Wu_, Feb 16 2022