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A357749 Sorted list of nonzero numbers x, y, z that occur in solutions to the equation (x + y)^2 + (y + z)^2 + (z + x)^2 = 12*x*y*z. 2

%I #11 Oct 18 2022 13:50:53

%S 1,3,13,61,217,291,1393,3673,4683,6673,16693,31971,62221,106153,

%T 153181,360517,733933,1054081,1285131,1709221,2430493,3516483,4778353,

%U 16848481,17857153,21717363,27755113,38745493,55764867,80725921,98938381,185236633,302517517,386781123

%N Sorted list of nonzero numbers x, y, z that occur in solutions to the equation (x + y)^2 + (y + z)^2 + (z + x)^2 = 12*x*y*z.

%H 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], 2022.

%e The first few solutions are (x, y, z) = (1, 1, 1), (1, 1, 3), (1, 3, 13), (1, 13, 61), (1, 61, 291), (1, 291, 1393), (1, 1393, 6673), (1, 6673, 31971), (3, 13, 217), (3, 217, 3673), ..., so 1, 3, 13, 61, 217, 291, ... are terms.

%o (PARI) a357749(steps) = {L=List(); listput(L,[1,1,1]); listput(L,[1,1,3]); listput(L,[1,13,3]); for(n=1,steps,my(mp,mv);for(l=1,#L,mv=vecmax(L[l],&mp); my (a=L[l][1], b=L[l][2], c=L[l][3], s=a+b+c); if(mp==1,listput(L,[a,6*a*c-s,c]);listput(L,[a,b,6*a*b-s])); if(mp==2,listput(L,[6*b*c-s,b,c]);listput(L,[a,b,6*a*b-s])); if(mp==3,listput(L,[6*b*c-s,b,c]);listput(L,[a,6*a*c-s,c]))));M=List(); for (k=1, #L, for(j=1, 3, listput(M,L[k][j]))); vecsort(M,,8)};

%o a357749(13)[1..35]

%Y Cf. A002559, A101368, A357870.

%K nonn

%O 1,2

%A _Hugo Pfoertner_, Oct 18 2022

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Last modified March 29 09:28 EDT 2024. Contains 371268 sequences. (Running on oeis4.)