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A291694
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Array of Markov triples (x,y,z) sorted by z, read by rows.
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2
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1, 1, 1, 1, 1, 2, 1, 2, 5, 1, 5, 13, 2, 5, 29, 1, 13, 34, 1, 34, 89, 2, 29, 169, 5, 13, 194, 1, 89, 233, 5, 29, 433, 1, 233, 610, 2, 169, 985, 13, 34, 1325, 1, 610, 1597, 5, 194, 2897, 1, 1597, 4181, 2, 985, 5741, 5, 433, 6466, 13, 194, 7561, 34, 89, 9077, 1, 4181, 10946, 29, 169, 14701
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OFFSET
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1,6
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COMMENTS
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The positive integers x, y, z satisfy the Diophantine equation x^2 + y^2 + z^2 = 3*x*y*z, 1 <= x <= y <= z.
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.31.3 Markov-Hurwitz Equation, p. 200.
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LINKS
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EXAMPLE
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The array of Markov triples begins:
(1, 1, 1),
(1, 1, 2),
(1, 2, 5),
(1, 5, 13),
(2, 5, 29),
(1, 13, 34),
...
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MATHEMATICA
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triples = 30; depth0 = 10 (* adjust depth according to message after first run *) ; Clear[zz, fx, fy]; fx[1] = fy[1] = fx[2] = fy[2] = fx[5] = 1;
fy[5] = 2; zz[n_] := zz[n] = Module[{f, x, y, z}, f[] = {1, 2, 5}; f[ud___, u(*up*)] := f[ud, u] = Module[{g = f[ud]}, x = g[[1]]; y = g[[3]]; z = 3*g[[1]]*g[[3]] - g[[2]]; fx[z] = x; fy[z] = y; {x, y, z}]; f[ud___, d(*down*)] := f[ud, d] = Module[{g = f[ud]}, x = g[[2]]; y = g[[3]]; z = 3*g[[2]]*g[[3]] - g[[1]]; fx[z] = x; fy[z] = y; {x, y, z}]; f @@@ Tuples[{u, d}, n] // Flatten // Union // PadRight[#, triples]&]; zz[n = depth0]; zz[n++]; While[zz[n] != zz[n - 1], n++]; Print["depth = n = ", n]; xyz = {fx[#], fy[#], #} & /@ zz[n]; Flatten[xyz]
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PROG
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(PARI) N=5000;
for(k=1, N, for(j=1, k, for(i=1, j, if(i*j>k, break); if(i^2+j^2+k^2==3*i*j*k, print1(i, ", ", j, ", ", k, ", "))))); \\ Seiichi Manyama, Feb 16 2022
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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