A245325 Numerators of an enumeration system of the reduced nonnegative rational numbers.

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

Offset

1,3

Comments

a(n)/A245326(n) enumerates all the reduced nonnegative rational numbers exactly once.

If the terms (n>0) are written as an array (in a left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...

1,

1,2,

2,1,3,3,

3,3,2,1,5,4,5,4,

5,4,5,4,3,3,2,1,8,7,7,5,8,7,7,5,

8,7,7,5,8,7,7,5,5,4,5,4,3,3,2,1,13,11,12,9,11,10,9,6,13,11,12,9,11,10,9,6,

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,

1,2,

2, 1,3,3,

3, 3, 2,1, 5, 4,5,4,

5, 4, 5,4, 3, 3,2,1, 8, 7, 7,5, 8, 7,7,5,

8,7,7,5,8,7,7,5,5,4,5,4,3,3,2,1,13,11,12,9,11,10,9,6,13,11,12,9,11,10,9,6,

then each column is an arithmetic sequence. The differences of the arithmetic sequences give the sequence A071585 (a(2^(m+1)-1-k) - a(2^m-1-k) = A071585(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 the Stern-Brocot sequence), and, more precisely, the reverses of blocks of A229742 (a(2^m+k) = A229742(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 A245327.

Links

Table of n, a(n) for n=1..100.

Index entries for fraction trees

Formula

a(n) = A002487(A059893(A180200(n))) = A002487(1+A059893(A154435(n))). - Yosu Yurramendi, Sep 20 2021

Prog

(R)
blocklevel <- 6 # arbitrary
a <- 1
for(m in 0:blocklevel) for(k in 0:(2^(m-1)-1)){
  a[2^(m+1)+k]             <- a[2^m+2^(m-1)+k]
  a[2^(m+1)+2^(m-1)+k]     <- a[2^m+k]
  a[2^(m+1)+2^m+k]         <- a[2^(m+1)+k] +  a[2^m+k]
  a[2^(m+1)+2^m+2^(m-1)+k] <- a[2^(m+1)+2^m+k]
}
a

Crossrefs

Cf. A245326, A002487, A071585, A229742, A273494.

Sequence in context: A164999 A292030 A162909 * A091224 A308684 A112182

Adjacent sequences: A245322 A245323 A245324 * A245326 A245327 A245328

Keyword

nonn,frac,changed

Author

Yosu Yurramendi, Jul 18 2014

Status

approved