%I A002559 M1432 N0566
%S A002559 1,2,5,13,29,34,89,169,194,233,433,610,985,1325,1597,2897,4181,5741,
%T A002559 6466,7561,9077,10946,14701,28657,33461,37666,43261,51641,62210,75025,
%U A002559 96557,135137,195025,196418,294685,426389,499393,514229,646018,925765
%N A002559 Markoff (or Markov) numbers: union of numbers x, y, z satisfying x^2 
               + y^2 + z^2 = 3xyz.
%C A002559 A004280 gives indices of Fibonacci numbers (A000045) which are also Markoff 
               (or Markov) numbers.
%C A002559 As mentioned by Conway and Guy, all odd-index Pell numbers (A001653) 
               also appear in this sequence. The positions of the Fibonacci and 
               Pell numbers in this sequence are given in A158381 and A158384, respectively. 
               - T. D. Noe, Mar 19 2009
%C A002559 Assuming that each solution (x,y,z) is ordered x <= y <= z, the open 
               problem is to prove that each z value occurs only once. There are 
               no counterexamples in the first 1046858 terms, which have z values 
               < Fibonacci(5001) = 6.2763...*10^1044. - T. D. Noe, Mar 19 2009
%D A002559 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002559 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002559 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 
               1996, p. 187.
%D A002559 R. Descombes, Problemes d'approximation diophantienne. Enseignement Math. 
               (2) 6 1960 18-26.
%D A002559 R. K. Guy, Don't try to solve these problems, Amer. Math. Monthly, 90 
               (1983), 35-41.
%D A002559 R. K. Guy, Unsolved Problems in Number Theory, D12.
%D A002559 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, notes on ch. 24.6 (p. 412)
%H A002559 T. D. Noe, <a href="b002559.txt">Table of n, a(n) for n=1..1000</a>
%H A002559 T. Ace, <a href="http://www.minortriad.com/markov.html">Markov numbers</
               a>
%H A002559 M. L. Lang & S. P. Tan, <a href="http://arXiv.org/abs/math.NT/0508443">
               A simple proof of the Markoff conjecture for prime powers</a>
%H A002559 M. L. Lang & S. P. Tan, <a href="http://www.math.nus.edu.sg/~mattansp/
               markoffconjectureprimepower.pdf">A Simple Proof Of The Markoff Conjecture 
               For Prime Powers</a>
%H A002559 J. Propp, <a href="http://www.math.wisc.edu/~propp/markoff-talk.html">
               The combinatorics of Markov numbers</a>
%H A002559 M. Waldschmidt, <a href="http://arXiv.org/abs/math.NT/0312440">Open Diophantine 
               problems</a>
%H A002559 Y. Zhang, <a href="http://arXiv.org/abs/math.NT/0606283">An Elementary 
               Proof of Markoff Conjecture for Prime Powers</a>
%t A002559 m = {1}; Do[x = m[[i]]; y = m[[j]]; a = (3*x*y + Sqrt[ -4*x^2 - 4*y^2 
               + 9*x^2*y^2])/2; b = (3*x*y + Sqrt[ -4*x^2 - 4*y^2 + 9*x^2*y^2])/
               2; If[ IntegerQ[a], m = Union[ Join[m, {a}]]]; If[ IntegerQ[b], m 
               = Union[ Join[m, {b}]]], {n, 8}, {i, Length[m]}, {j, i}]; Take[m, 
               40] (from Robert G. Wilson v (rgwv(at)rgwv.com), Oct 05 2005)
%Y A002559 Sequence in context: A126656 A026522 A122491 this_sequence A049097 A045366 
               A158708
%Y A002559 Adjacent sequences: A002556 A002557 A002558 this_sequence A002560 A002561 
               A002562
%K A002559 nonn,nice,easy
%O A002559 1,2
%A A002559 N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)