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%I #24 Feb 10 2024 09:36:57
%S 1,2,5,29,13,433,169,194,34,37666,6466,14701,985,7561,2897,1325,89,
%T 48928105,3276509,8399329,96557,7453378,1278818,499393,5741,4400489,
%U 294685,1686049,43261,135137,51641,9077,233,5528778008357,63557570729,370238963953,285018617,162930183509,10910721905,1873012681
%N 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.
%C The row length is r(-2) = r(-1) = 1 and r(L) = 2^L, for L >= 0.
%C For this Markoff tree MTree (with increasingly ordered members of the triples) see the Zagier link, FIGURE 2.
%C 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).
%C 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).
%C Here only the tree levels with the maximal members of the Markoff triples are recorded, and this tree is called MTreemax.
%C 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.
%C 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.
%C The rightmost entries are the odd-indexed Fibonacci numbers: T(L, r(L)) = F(2*L + 5) = A001519(2*L + 5), l >= -2.
%H John Tyler Rascoe, <a href="/A327345/b327345.txt">Rows n = -2..11, flattened</a>
%H Don Zagier, <a href="http://dx.doi.org/10.1090/S0025-5718-1982-0669663-7">On the number of Markoff numbers below a given bound</a>, Mathematics of Computation 39:160 (1982), pp. 709-723.
%e The MTreemax begins (vertical bars, semicolons, and commas indicate the binary structure, for L >= 0):
%e L= -2: [1]
%e L= -1: [2]
%e L= 0: [5]
%e L= 1: [29 || 13]
%e L= 2: [433 169 || 194 34]
%e L= 3: [37666 6466 | 14701 985 || 7561 2897 | 1325 89]
%e L= 4: [48928105 3276509 ; 8399329 96557 | 7453378 1278818 ; 499393 5741 || 4400489 294685 ; 1686049 43261 | 135137 51641 ; 9077 233]
%e 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]
%e ...
%e -----------------------------------------------------------------------------
%e Left and Right rules in the MT:
%e 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).
%e T(2, 1) = 433 comes from the Right rule applied to MT(1, 1) -> (5, 29, 3*5*29 - 2) = (5, 29, 433).
%e -----------------------------------------------------------------------------
%o (Python)
%o def Mtree(a):
%o left = tuple(sorted([a[0],a[2],(3*a[0]*a[2])-a[1]]))
%o right = tuple(sorted([a[1],a[2],(3*a[1]*a[2])-a[0]]))
%o if left == right: return(left,)
%o else: return(max(left,right),min(left,right))
%o def A327345_rowlist(maxrow):
%o A,B,S = [[(1,1,1)]],[],set()
%o for n in range(0,maxrow+3):
%o A.append([])
%o for j in A[n]:
%o S.add(j)
%o B.append(max(j))
%o for k in Mtree(j):
%o if k not in S:
%o A[n+1].append(k)
%o return(B) # _John Tyler Rascoe_, Feb 10 2024
%Y Cf. A002519, A002559.
%K nonn,tabf,look,easy
%O -2,2
%A _Wolfdieter Lang_, Sep 13 2019
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