

A327345


Irregular triangle read by rows: Maximal numbers of the Markoff triples at level L of the Markoff tree, with members of the triples ordered increasingly.


0



1, 2, 5, 29, 13, 433, 169, 194, 34, 37666, 6466, 14701, 985, 7561, 2897, 1325, 89, 48928105, 3276509, 8399329, 96557, 7453378, 1278818, 499393, 5741, 4400489, 294685, 1686049, 43261, 135137, 51641, 9077, 233, 5528778008357, 63557570729, 370238963953, 285018617, 162930183509, 10910721905, 1873012681
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OFFSET

2,2


COMMENTS

The row length is r(2) = r(1) = 1 and r(L) = 2^L, for L >= 0.
For this Markoff tree MTree (with increasingly ordered members of the triples) see the Zagier link, FIGURE 2.
The levels MTree(L), for L >= 2 have r(L) nodes. The root node of the (completely) binary tree is at level L = 0 with triple (1, 2, 5).
The rule for the left successors (going from top to bottom) in MTree is Left: (x, y, m) > (x, m, 3*x*m  y), and for the right successor it is Right: (x, y, m) > (y, m, 3*y*m  x).
Here only the tree levels with the maximal members of the Markoff triples are recorded, and this tree is called MTreemax.
Each member of A002559 (Markoff numbers, sorted increasingly) appears in the Markoff tree as maximal member of some triple, hence every member of A002559 appears in the present tree MTreemax.
The FrobeniusMarkoff uniqueness conjecture is: each member of A002559 appears precisely once as maximal number of some triple in MT. Hence it is the conjectured that in MTreemax each member of A002559 appears only once.
The rightmost entries are the oddindexed Fibonacci numbers: T(L, r(L)) = F(2*L + 5) = A001519(2*L + 5), l >= 2.


LINKS

Table of n, a(n) for n=2..37.
Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709723.


EXAMPLE

The MTreemax begins (vertical bars, semicolons, and commas indicate the binary structure, for L >= 0):
L= 2: [1]
L= 1: [2]
L= 0: [5]
L= 1: [29  13]
L= 2: [433 169  194 34]
L= 3: [37666 6466  14701 985  7561 2897  1325 89]
L= 4: [48928105 3276509 ; 8399329 96557  7453378 1278818 ; 499393 5741  4400489 294685 ; 1686049 43261  135137 51641 ; 9077 233]
L = 5: [5528778008357 63557570729 , 370238963953 285018617 ; 162930183509 10910721905 , 1873012681 1441889  328716329765 3778847945 , 56399710225 111242465 ; 1475706146 253191266 , 16964653 33461  99816291793 2561077037 6684339842 11485154 14653451665 981277621 375981346 646018  537169541 13782649 , 205272962 2012674 ; 2423525 925765 , 62210 610]
...

Left and Right rules in the MT:
T(2, 2) = 169 comes from the Left rule applied to the triple with maximum 29 which is MT(1, 1) = (2, 5, 29)  > (2, 29, 3*2*29  5) = (2, 29, 169) = MT(2, 2).
T(2, 1) = 433 comes from the Right rule applied to MT(1, 1) > (5, 29, 3*5*29  2) = (5, 29, 433).



CROSSREFS

Cf. A002519, A002559.
Sequence in context: A000472 A248235 A358444 * A049050 A344020 A178322
Adjacent sequences: A327342 A327343 A327344 * A327346 A327347 A327348


KEYWORD

nonn,tabf,easy


AUTHOR

Wolfdieter Lang, Sep 13 2019


STATUS

approved



