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.

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

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 Frobenius-Markoff 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 odd-indexed Fibonacci numbers: T(L, r(L)) = F(2*L + 5) = A001519(2*L + 5), l >= -2.

Links

John Tyler Rascoe, Rows n = -2..11, flattened

Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709-723.

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).
-----------------------------------------------------------------------------

Prog

(Python)
def Mtree(a):
    left = tuple(sorted([a[0], a[2], (3*a[0]*a[2])-a[1]]))
    right = tuple(sorted([a[1], a[2], (3*a[1]*a[2])-a[0]]))
    if left == right: return(left, )
    else: return(max(left, right), min(left, right))
def A327345_rowlist(maxrow):
    A, B, S = [[(1, 1, 1)]], [], set()
    for n in range(0, maxrow+3):
        A.append([])
        for j in A[n]:
            S.add(j)
            B.append(max(j))
            for k in Mtree(j):
                if k not in S:
                    A[n+1].append(k)
    return(B) # John Tyler Rascoe, Feb 10 2024

Crossrefs

Cf. A002519, A002559.

Sequence in context: A000472 A248235 A358444 * A049050 A344020 A178322

Adjacent sequences: A327342 A327343 A327344 * A327346 A327347 A327348

Keyword

nonn,tabf,look,easy

Author

Wolfdieter Lang, Sep 13 2019

Status

approved