cp's OEIS Frontend

A368134 Characteristic numbers of Markov triples in the binary tree A368546.

%I A368134 #21 Sep 22 2024 15:20:36
%S A368134 2,5,12,13,75,179,70,34,507,2923,1120,2673,15571,6089,408,89,3468,
%T A368134 51709,19760,113922,1701181,651838,16725,39916,3472225,20226717,
%U A368134 1354498,529673,3087111,206855,2378,233,23763,925943,353702,5273811,205543262,78545995,770133
%N A368134 Characteristic numbers of Markov triples in the binary tree A368546.
%C A368134 The characteristic number u of a Markov triple (r, m, s) is the solution in (0, m) of r * x == s (mod m). It satisfies u^2 == -1 (mod m), so that v = (u^2 + 1) / m is also an integer. The other solution in (0,m) of u^2 == -1 (mod m), namely m - u, is always greater than u, so u < m / 2.
%C A368134 The Markov tree may be formulated in terms of a set of Cohn matrices. There is a one-parameter family of such sets, parametrized by an integer c. Given a vertex of the Markov tree with Farey triple (x, y, z) and Markov triple (r, m, s), producing characteristic number u and v = (u^2 + 1) / m, the Cohn matrix C_y(c) with parameter c is
%C A368134    [               c * m + u                                       m      ]
%C A368134    [(3 * c - c^2) * m - (2 * c - 3) * u - v                (3 - c) * m - u].
%C A368134   Then the vertex is associated with a triple of Cohn matrices, (R, R S, S), where R = C_x(c), RS = C_y(c), and S = C_z(c). See A368546 for a description of Farey and Markov triples. The left child of the vertex is associated with the triple (R, R^2 S, RS) and the right child with (RS, R S^2, S).
%D A368134 Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013. x+257 pp. ISBN: 978-3-319-00887-5; 978-3-319-00888-2 MR3098784
%H A368134 Martin Aigner, <a href="https://archive.org/details/markovstheorem100000aign">Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings</a>, [archive.org copy of the book]
%F A368134 Recurrence: The left child of the Markov triple (r, m, s) is (r, 3rm - s, m); the right child is (m, 3ms - r, s). The corresponding triple of characteristic numbers (t, u, v) has left child (t, 3ru - v, u) and right child (u, 3us - t, v). Initial Markov triple: (1, 5, 2), initial characteristic number triple: (0, 2, 1).
%e A368134 The initial rows of the binary tree are
%e A368134                                   2
%e A368134              5                                       12
%e A368134    13                 75                    179                70
%e A368134 34    507        2923   1120            2673  15571       6089   408
%o A368134 (SageMath)
%o A368134 rowM = [[1,5,2]]
%o A368134 rowU = [[0,2,1]]
%o A368134 a368134 = [2]
%o A368134 for rw in range(1,6):
%o A368134     prevRowM = rowM
%o A368134     prevRowU = rowU
%o A368134     rowM = []
%o A368134     rowU = []
%o A368134     for i in range(len(prevRowM)):
%o A368134         [r,m,s] = prevRowM[i]
%o A368134         [t,u,v] = prevRowU[i]
%o A368134         ltM = [r,3*r*m - s,m]
%o A368134         rtM = [m,3*m*s - r,s]
%o A368134         ltU = [t,3*r*u - v,u]
%o A368134         rtU = [u,3*u*s - t,v]
%o A368134         rowM = rowM + [ltM,rtM]
%o A368134         rowU = rowU + [ltU,rtU]
%o A368134         a368134 = a368134 + [ltU[1],rtU[1]]
%o A368134 a368134
%Y A368134 Cf. A368546.
%K A368134 nonn,tabf
%O A368134 0,1
%A A368134 _William P. Orrick_, Jan 11 2024