This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A253809 #12 Jan 22 2020 07:16:25 %S A253809 1,1,1,1,1,2,1,5,2,5,1,13,1,34,2,29,5,13,1,89,5,29,1,233,2,169,13,34, %T A253809 1,610,5,194,1,1597,2,985,5,433,13,194,34,89,1,4181,29,169,1,10946,2, %U A253809 5741,29,433,5,2897,13,1325,89,233,1,28657 %N A253809 Array of pairs (x,y) of Markoff triples (x,y,z) with x <= y <= z, for z given in A002559. %C A253809 Frobenius' conjecture on Markoff triples is that the maximal member z of the triple of positive integers (x,y,z), satisfying x^2 + y^2 + z^2 - 3*x*y*z = 0, with x <= y <= z, determines x and y uniquely. Also, each entry from A002559 (Markoff numbers) is conjectured to appear as a maximal member z. If an entry A002559(n) should not appear as z then one puts z(n) = 0 and row n will be 0, 0. %C A253809 If this Frobenius conjecture is true then the row length of this array is always 2, and only positive numbers appear. %D A253809 R. A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125. %D A253809 See also A002559. %H A253809 Feng-Juan Chen and Yong-Gao Chen, <a href="https://doi.org/10.1016/j.jnt.2012.12.018">On the Frobenius conjecture for Markoff numbers</a>, J. Number Theory 133 (2013) 2363-2373. %H A253809 Don Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2007348/fulltext.pdf">On the number of Markoff numbers below a given bound</a>, Mathematics of Computation 39:160 (1982), pp. 709-723. %H A253809 See also A002559. %e A253809 The array A(n,k) begins: %e A253809 If the Frobenius conjecture is true there will only be one pair x(1,n), y(1,n) for each z(n). %e A253809 n z(n) \ k=1: x(1,n) k=2: y(1,n) ... %e A253809 1 1: 1 1 %e A253809 2 2: 1 1 %e A253809 3 5: 1 2 %e A253809 4 13: 1 5 %e A253809 5 29: 2 5 %e A253809 6 34: 1 13 %e A253809 7 89: 1 34 %e A253809 8 169: 2 29 %e A253809 9 194: 5 13 %e A253809 10 233: 1 89 %e A253809 11 433: 5 29 %e A253809 12 610: 1 233 %e A253809 13 985: 2 169 %e A253809 14 1325: 13 34 %e A253809 15 1597: 1 610 %e A253809 16 2897: 5 194 %e A253809 17 4181: 1 1597 %e A253809 18 5741: 2 985 %e A253809 19 6466: 5 433 %e A253809 20 7561: 13 194 %e A253809 21 9077: 34 89 %e A253809 22 10946: 1 4181 %e A253809 23 14701: 29 169 %e A253809 24 28657: 1 10946 %e A253809 25 33461: 2 5741 %e A253809 26 37666: 29 433 %e A253809 27 43261: 5 2897 %e A253809 28 51641: 13 1325 %e A253809 29 62210: 89 233 %e A253809 30 75025: 1 28657 %e A253809 ... %Y A253809 Cf. A002559. %K A253809 nonn,tabf %O A253809 1,6 %A A253809 _Wolfdieter Lang_, Jan 28 2015