A291694 Array of Markov triples (x,y,z) sorted by z, read by rows.

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

Offset: 1

Author

Jean-François Alcover, Aug 30 2017

Keywords

• nonn
• tabf

Comments

The positive integers x, y, z satisfy the Diophantine equation x^2 + y^2 + z^2 = 3*x*y*z, 1 <= x <= y <= z.

Examples

			The array of Markov triples begins:
  (1,  1,  1),
  (1,  1,  2),
  (1,  2,  5),
  (1,  5, 13),
  (2,  5, 29),
  (1, 13, 34),
  ...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.31.3 Markov-Hurwitz Equation, p. 200.

Links

• Eric Weisstein's World of Mathematics, Markov Number.
• Wikipedia, Markov number.

Crossrefs

Cf. A002559 (main entry for this sequence), A178444, A256395, A261613, A351372.

Programs

Mathematica

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]

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