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 A245328 Denominators in recursive bijection from positive integers to positive rationals (the bijection is f(1) = 1, f(2n) = 1/(f(n)+1), f(2n+1) = f(n)+1). 8
 1, 2, 1, 3, 2, 3, 1, 5, 3, 5, 2, 4, 3, 4, 1, 8, 5, 8, 3, 7, 5, 7, 2, 7, 4, 7, 3, 5, 4, 5, 1, 13, 8, 13, 5, 11, 8, 11, 3, 12, 7, 12, 5, 9, 7, 9, 2, 11, 7, 11, 4, 10, 7, 10, 3, 9, 5, 9, 4, 6, 5, 6, 1, 21, 13, 21, 8, 18, 13, 18, 5, 19, 11, 19, 8, 14, 11, 14, 3, 19, 12, 19, 7, 17, 12, 17, 5, 16, 9, 16, 7, 11, 9, 11, 2, 18, 11, 18, 7, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A.....(n)/a(n) enumerates all the reduced nonnegative rational numbers exactly once. 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, 2,1, 3,2, 3,1, 5,3, 5,2, 4,3, 4,1, 8,5, 8,3, 7,5, 7,2, 7,4, 7,3,5,4,5,1, 13,8,13,5,11,8,11,3,12,7,12,5,9,7,9,2,11,7,11,4,10,7,10,3,9,5,9,4,6,5,6,1, then the sum of the m-th row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence. If the rows are written in a right-aligned fashion:                                                                         1,                                                                       2,1,                                                                   3,2,3,1,                                                           5,3,5,2,4,3,4,1,                                        8,5, 8,3, 7,5, 7,2,7,4,7,3,5,4,5,1, 13,8,13,5,11,8,11,3,12,7,12,5,9,7,9,2,11,7,11,4,10,7,10,3,9,5,9,4,6,5,6,1, then each column is an arithmetic sequence. The differences of the arithmetic sequences, except the first on the right, give the sequence A093873 (Numerators in Kepler's tree of harmonic fractions) (a(2^(m+1)-1-k) - a(2^m-1-k) = A093873(k), m = 0,1,2,..., k = 0,1,2,...,2^m-1). If the sequence is considered by blocks of length 2^m, m = 0,1,2,..., the blocks of this sequence are permutations of terms of blocks from A002487 (Stern's diatomic series or Stern-Brocot sequence), and, more precisely, the reverses of blocks of A020651 ( a(2^m+k) = A020651(2^(m+1)-1-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1). Moreover, each block is the bit-reversed permutation of the corresponding block of A245326. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..16383, rows 1-14, flattened. FORMULA a(2n) = A245327(2n+1) , a(2n+1) = A245328(2n) , n=1,2,3,... a((2*n+1)*2^m - 1) = A273493(n), n > 0, m >= 0. For n = 0 A273493(0) = 1 is needed. - Yosu Yurramendi, Mar 02 2017 MATHEMATICA f[n_] := Which[n == 1, 1, EvenQ@ n, 1/(f[n/2] + 1), True, f[(n - 1)/2] + 1]; Table[Denominator@ f@ k, {n, 7}, {k, 2^(n - 1), 2^n - 1}] // Flatten (* Michael De Vlieger, Mar 02 2017 *) PROG (R) N <- 25 # arbitrary a <- c(1, 2, 1) for(n in 1:N){   a[4*n]   <- a[2*n] + a[2*n+1]   a[4*n+1] <- a[2*n]   a[4*n+2] <- a[2*n] + a[2*n+1]   a[4*n+3] <-          a[2*n+1] } a CROSSREFS Cf. A002487, A020651, A093873, A245326, A245327, A273493. Sequence in context: A002487 A318509 A263017 * A060162 A026730 A318691 Adjacent sequences:  A245325 A245326 A245327 * A245329 A245330 A245331 KEYWORD nonn,frac AUTHOR Yosu Yurramendi, Jul 18 2014 STATUS approved

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Last modified October 21 06:39 EDT 2019. Contains 328292 sequences. (Running on oeis4.)