A245326 Denominators of an enumeration system of the reduced nonnegative rational numbers.

1, 2, 1, 3, 3, 2, 1, 5, 4, 5, 4, 3, 3, 2, 1, 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, 8, 7, 7, 5, 8, 7, 7, 5, 5, 4, 5, 4, 3, 3, 2, 1, 21, 18, 19, 14, 19, 17, 16, 11, 18, 15, 17, 13, 14, 13, 11, 7, 21, 18, 19, 14, 19, 17, 16, 11, 18, 15, 17, 13, 14, 13, 11, 7, 13, 11, 12, 9, 11

Offset: 1

Author

Yosu Yurramendi, Jul 18 2014

Keywords

• nonn
• frac

Comments

A245325(n)/a(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,

2, 1,

3, 3, 2,1,

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

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

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

then the sum of the m-th row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence. These Fibonacci sequences are equal to Fibonacci sequences from A...... except for the first terms of those sequences.

If the rows are written in a right-aligned fashion:

1,

2,1,

3,3,2,1,

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

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

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

then each column is constant and the terms are from A071585 (a(2^m-1-k) = A071585(k), k = 0,1,2,...).

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 A071766 (a(2^m+k) = A071766(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 A245328.

Links

• Index entries for fraction trees

Crossrefs

Cf. A245325, A002487, A071585, A071766, A273494.

Cf. A002487, A059893, A154435.

Programs

PARI

a(n) = my(A=1); for(i=0, logint(n, 2), if(bittest(2*n, i), A++, A=(A+1)/A)); denominator(A) \\ Mikhail Kurkov, Feb 20 2023

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+k] + a[2^m+2^(m-1)+k]
  a[2^(m+1)+2^(m-1)+k]     <- a[2^(m+1)+k]
  a[2^(m+1)+2^m+k]         <- a[2^m+k]
  a[2^(m+1)+2^m+2^(m-1)+k] <- a[2^m+2^(m-1)+k]
}
a

Formula

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