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 A162911 Numerators of drib tree fractions, where drib is the bit-reversal permutation tree of the Bird tree. 7
 1, 1, 2, 2, 3, 1, 3, 3, 5, 1, 4, 3, 4, 2, 5, 5, 8, 2, 7, 4, 5, 3, 7, 4, 7, 1, 5, 5, 7, 3, 8, 8, 13, 3, 11, 7, 9, 5, 12, 5, 9, 1, 6, 7, 10, 4, 11, 7, 11, 3, 10, 5, 6, 4, 9, 7, 12, 2, 9, 8, 11, 5, 13, 13, 21, 5, 18, 11, 14, 8, 19, 9, 16, 2, 11, 12, 17, 7, 19, 9, 14, 4, 13, 6, 7, 5, 11, 10, 17, 3, 13 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The drib tree is an infinite binary tree labeled with rational numbers. It is generated by the following iterative process: start with the rational 1; for the left subtree increment and then reciprocalize the current rational; for the right subtree interchange the order of the two steps: the rational is first reciprocalized and then incremented. Like the Stern-Brocot and the Bird tree, the drib tree enumerates all the positive rationals (A162911(n)/A162912(n)). From Yosu Yurramendi, Jul 11 2014: (Start) If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,... 1, 1, 2, 2, 3,1, 3, 3, 5,1, 4, 3, 4,2, 5, 5, 8,2, 7, 4, 5,3, 7,4, 7,1, 5, 5, 7,3, 8, 8,13,3,11, 7, 9,5,12,5, 9,1, 6, 7,10,4,11,7,11,3,10,5,6,4, 9, 7,12,2, 9, 8,11,5,13, 13,21,5,18,11,14,8 19,9,16,2,11,12 17,7,19,9,14,4,13,6,7,5,11,10,17,3,13,11,15,7,18, ... then the sum of the m-th row is 3^m (m = 0,1,2,), each column k is a Fibonacci-type sequence. If the rows are written in a right-aligned fashion:                                                                                   1,                                                                                1, 2,                                                                           2, 3,1, 3,                                                                3, 5,1, 4, 3, 4,2, 5,                                           5, 8,2, 7,4, 5,3, 7, 4, 7,1, 5, 5, 7,3, 8, 8,13,3,11, 7, 9,5,12,5, 9,1,6,7,10,4,11, 7,11,3,10,5, 6,4, 9, 7,12,2, 9, 8,11,5,13, ... 11,18,4,15,10,13,7,17,6,11,1,7,9,13,5,14,12,19,5 17,9,11,7,16 11,19,3,14,13,18,8,21, then each column k also is a Fibonacci-type sequence. If the sequence is considered by blocks of length 2^m, m = 0,1,2,..., the blocks of this sequence are the reverses of blocks of A162912 (a(2^m+k) = A162912(2^(m+1)-1-k), m = 0,1,2,..., k = 0..2^m-1). (End) From Yosu Yurramendi, Jan 12 2017: (Start) a(2^(m+2m'  )     + A020988(m'))   = A000045(m+1), m>=0, m'>=0 a(2^(m+2m'+1)     + A020989(m'))   = A000045(m+3), m>=0, m'>=0 a(2^(m+2m'  ) - 1 - A002450(m'))   = A000045(m+1), m>=0, m'>=0 a(2^(m+2m'+1) - 1 - A072197(m'-1)) = A000045(m+3), m>=0, m'>0 a(2^(m+1) -1) = A000045(m+2), m>=0. (End) LINKS R. Hinze, Functional pearls: the bird tree, J. Funct. Programming 19 (2009), no. 5, 491-508. FORMULA a(n) where a(1) = 1; a(2n) = b(n); a(2n+1) = a(n) + b(n); and b(1) = 1; b(2n) = a(n) + b(n); b(2n+1) = a(n). a(2^(m+1)+2*k) = a(2^(m+1)-k-1), a(2^(m+1)+2*k+1) = a(2^(m+1)-k-1) + a(2^m+k), a(1) = 1, m>=0, k=0..2^m-1. - Yosu Yurramendi, Jul 11 2014 a(2^(m+1) + 2*k) = A162912(2^m + k), m >= 0, 0 <= k < 2^m. a(2^(m+1) + 2*k + 1) = a(2^m + k) + A162912(2^m + k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 30 2016 a(n*2^m + A176965(n)) = A268087(n), n > 0, m >= 0. For n = 0 A086592(0) = 1 is needed. - Yosu Yurramendi, Feb 20 2017 EXAMPLE The first four levels of the drib tree: [1/1], [1/2, 2/1], [2/3, 3/1, 1/3, 3/2], [3/5, 5/2, 1/4, 4/3, 3/4, 4/1, 2/5, 5/3]. PROG (Haskell) import Ratio; drib :: [Rational]; drib = 1 : map (recip . succ) drib \/ map (succ . recip) drib; (a : as) \/ bs = a : (bs \/ as); a162911 = map numerator drib; a162912 = map denominator drib (R) blocklevel <- 6 # arbitrary a <- 1 for(m in 0:blocklevel) for(k in 0:(2^m-1)){   a[2^(m+1)+2*k  ] <- a[2^(m+1)-1-k]   a[2^(m+1)+2*k+1] <- a[2^(m+1)-1-k] + a[2^m+k] } a # Yosu Yurramendi, Jul 11 2014 CROSSREFS This sequence is the composition of A162909 and A059893: a(n) = A162909(A059893(n)). This sequence is a permutation of A002487(n+1). Sequence in context: A224764 A077268 A088741 * A245327 A131821 A204123 Adjacent sequences:  A162908 A162909 A162910 * A162912 A162913 A162914 KEYWORD easy,frac,nonn,changed AUTHOR Ralf Hinze (ralf.hinze(AT)comlab.ox.ac.uk), Aug 05 2009 STATUS approved

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