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 A368546 #64 Jun 09 2025 21:02:02 %S A368546 5,13,29,34,194,433,169,89,1325,7561,2897,6466,37666,14701,985,233, %T A368546 9077,135137,51641,294685,4400489,1686049,43261,96557,8399329, %U A368546 48928105,3276509,1278818,7453378,499393,5741,610,62210,2423525,925765,13782649,537169541 %N A368546 Alternative version of the Markov tree A327345. %C A368546 The Markov tree is a complete, infinite binary tree. Vertices are labeled by triples. The root vertex is (1, 5, 2). The left child of (a, b, c) is (a, 3*a*b - c, b); its right child is (b, 3*b*c - a, c). The sequence is a triangle read by rows consisting of the middle element of each triple, which is always the largest element of the triple. Row r contains 2^r elements. %C A368546 The tree contains contains exactly one representative of each class of permutation equivalent nonsingular solutions to Markov's equation, a^2 + b^2 + c^2 = 3 * a * b * c. Nonsingular solutions are those in which a, b, and c are three distinct numbers. The two singular triples (1, 1, 1) and (1, 2, 1) are omitted in this sequence. %C A368546 A consequence of Markov's equation is that the recurrence for the tree may be reformulated as follows: the left child of (a, b, c) is (a, (a^2 + b^2) / c, b); its right child is (b, (b^2 + c^2) / a, c). %C A368546 An open problem is to prove the uniqueness conjecture, which asserts that the largest element of a triple determines the other two. %C A368546 Frobenius proposed assigning a rational number index in (0,1) to each vertex of the tree, and hence to each term in this sequence. This is the Farey index, obtained by assigning the triple (0/1, 1/2, 1/1) to the root vertex and using the following rules to assign triples to the rest of the tree: the vertex labeled (u/v, w/x, y/z) with w = u + u and x = v + z has left child (u/v, (u+w)/(v+x), w/x) and right child (w/x, (w+y)/(x+z), y/z). The Farey index is the center element of the triple. Each rational number in (0, 1) appears as the Farey index of exactly one vertex of the tree. The index of a(n) is A007305(n+2) / A007306(n+2). %C A368546 A sequence of leftward steps in the tree produces odd-indexed Fibonacci numbers, A001519, which have Farey indices of the form 1 / n. A sequence of rightward steps in the tree produces odd-indexed Pell numbers, A001653, which have Farey indices of the form (n - 1) / n. A sequence of leftward steps followed by a single rightward step produces A350922, corresponding to Farey indices of the form 2 / (2 * n + 1). Alternating steps right, left, right, left, right, ... produces A064098, which corresponds to Farey indices of the form F(n) / F(n + 1), where F(n) is the n-th Fibonacci number. %D A368546 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 A368546 John Tyler Rascoe, <a href="/A368546/b368546.txt">Rows n = 0..10, flattened</a> %H A368546 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]. %H A368546 Robert A. Gore, <a href="https://arxiv.org/abs/2506.04299">Patterns Within the Markov Tree</a>, arXiv:2506.04299 [math.GM], 2025. %e A368546 The initial levels of the tree are as follows. (See p. 47 of Aigner's book.) %e A368546 (1,5,2) %e A368546 (1,13,5) (5,29,2) %e A368546 (1,34,13) (13,194,5) (5,433,29) (29,169,2) %e A368546 (1, (34, (13, (194, (5, (433, (29, (169, %e A368546 89, 1325, 7561, 2897, 6466, 37666, 14701, 985, %e A368546 ,34) 13) 194) 5) 433) 29) 169) 2) %o A368546 (SageMath) %o A368546 def stripUpToFirst1(w): %o A368546 x = w %o A368546 while x % 2 == 0: %o A368546 x = x // 2 %o A368546 return(x // 2) %o A368546 def stripUpToFirst0(w): %o A368546 x = w %o A368546 while x % 2 == 1: %o A368546 x = x // 2 %o A368546 if x == 0: %o A368546 return(None) %o A368546 else: %o A368546 return(x // 2) %o A368546 @CachedFunction %o A368546 def markovNumber(w): %o A368546 if w == None: %o A368546 return(2) %o A368546 elif w == 0: %o A368546 return(1) %o A368546 elif w == 1: %o A368546 return(5) %o A368546 elif w % 2 == 0: %o A368546 return(3*markovNumber(stripUpToFirst1(w))*markovNumber(w//2) - markovNumber(stripUpToFirst0(w//2))) %o A368546 else: %o A368546 return(3*markovNumber(stripUpToFirst0(w))*markovNumber(w//2) - markovNumber(stripUpToFirst1(w//2))) %o A368546 [markovNumber(w) for w in range(1,38)] %o A368546 (Python) %o A368546 def Mtree(x): return(x[0],(3*x[0]*x[1])-x[2],x[1]), (x[1],(3*x[1]*x[2])-x[0],x[2]) %o A368546 def A368546_rowlist(maxrow): %o A368546 A,B = [[(1,5,2)]],[] %o A368546 for n in range(maxrow+1): %o A368546 A.append([]) %o A368546 for j in A[n]: %o A368546 B.append(max(j)) %o A368546 for k in Mtree(j): %o A368546 A[n+1].append(k) %o A368546 return(B) # _John Tyler Rascoe_, Feb 09 2024 %Y A368546 Other presentations of the Markov numbers, Markov triples, or the Markov tree: A002559, A253809, A291694, A305313, A305314, A327345. %Y A368546 Subsequences in the Markov tree: A001519, A001653, A350922, A064098. %Y A368546 Farey tree: A007305, A007306. %K A368546 nonn,tabf %O A368546 0,1 %A A368546 _William P. Orrick_, Jan 04 2024