

A245327


Numerators 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 .


11



1, 1, 2, 2, 3, 1, 3, 3, 5, 2, 5, 3, 4, 1, 4, 5, 8, 3, 8, 5, 7, 2, 7, 4, 7, 3, 7, 4, 5, 1, 5, 8, 13, 5, 13, 8, 11, 3, 11, 7, 12, 5, 12, 7, 9, 2, 9, 7, 11, 4, 11, 7, 10, 3, 10, 5, 9, 4, 9, 5, 6, 1, 6, 13, 21, 8, 21, 13, 18, 5, 18, 11, 19, 8, 19, 11, 14, 3, 14, 12, 19, 7, 19, 12, 17, 5, 17, 9, 16, 7, 16, 9, 11, 2, 11, 11, 18, 7, 18, 11
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OFFSET

1,3


COMMENTS

a(n)/A245328(n) enumerates all the reduced nonnegative rational numbers exactly once.
If the terms (n>0) are written as an array (leftaligned fashion) with rows of length 2^m, m = 0,1,2,3,...
1,
1, 2,
2, 3,1, 3,
3, 5,2, 5,3, 4,1, 4,
5, 8,3, 8,5, 7,2, 7,4, 7,3, 7,4,5,1,5,
8,13,5,13,8,11,3,11,7,12,5,12,7,9,2,9,7,11,4,11,7,10,3,10,5,9,4,9,5,6,1,6,
then the sum of the mth row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence.
If the rows are written in a rightaligned fashion:
1,
1,2,
2,3,1,3,
3,5,2,5,3,4,1,4,
5, 8,3, 8,5, 7,2, 7,4,7,3,7,4,5,1,5,
8,13,5,13,8,11,3,11,7,12,5,12,7,9,2,9,7,11,4,11,7,10,3,10,5,9,4,9,5,6,1,6,
then each column is an arithmetic sequence.
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 SternBrocot sequence), and, more precisely, the reverses of blocks of A020650 ( a(2^m+k) = A020650(2^(m+1)1k), m = 0,1,2,..., k = 0,1,2,...,2^m1).
Moreover, each block is the bitreversed permutation of the corresponding block of A245325.


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..16383, rows 114, flattened.
Index entries for fraction trees


FORMULA

a(2n) = A245328(2n+1) , a(2n+1) = A245328(2n) , n=0,1,2,3,...
a((2*n+1)*2^m  2) = A273493(n), n > 0, m > 0. For n = 0, m > 0, A273493(0) = 1 is needed. For n = 1, m = 0, A273493(0) = 1 is needed. For n > 1, m = 0, numerator((2*n1) = num+den(n1).  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[Numerator@ 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, 1, 2)
for(n in 1:N){
a[4*n] < a[2*n+1]
a[4*n+1] < a[2*n] + a[2*n+1]
a[4*n+2] < a[2*n]
a[4*n+3] < a[2*n] + a[2*n+1]
}
a


CROSSREFS

Cf. A002487, A020651, A245325, A245328, A273493.
Sequence in context: A077268 A088741 A162911 * A131821 A204123 A237448
Adjacent sequences: A245324 A245325 A245326 * A245328 A245329 A245330


KEYWORD

nonn,frac


AUTHOR

Yosu Yurramendi, Jul 18 2014


STATUS

approved



