

A245326


Denominators of an enumeration system of the reduced nonnegative rational numbers


15



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
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OFFSET

1,2


COMMENTS

A245325(n)/a(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,
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 mth 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 the first terms of those sequences.
If the rows are written in a rightaligned 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^m1k) = 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 SternBrocot sequence), and, more precisely, the reverses of blocks of A071766 ( a(2^m+k) = A071766(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 A245328.


LINKS

Table of n, a(n) for n=1..100.
Index entries for fraction trees


PROG

(R)
blocklevel < 6 # arbitrary
a < 1
for(m in 0:blocklevel) for(k in 0:(2^(m1)1)){
a[2^(m+1)+k] < a[2^m+k] + a[2^m+2^(m1)+k]
a[2^(m+1)+2^(m1)+k] < a[2^(m+1)+k]
a[2^(m+1)+2^m+k] < a[2^m+k]
a[2^(m+1)+2^m+2^(m1)+k] < a[2^m+2^(m1)+k]
}
a


CROSSREFS

Cf. A245325, A002487, A071585, A071766, A273494.
Sequence in context: A324338 A047679 A179480 * A241534 A035050 A198790
Adjacent sequences: A245323 A245324 A245325 * A245327 A245328 A245329


KEYWORD

nonn,frac


AUTHOR

Yosu Yurramendi, Jul 18 2014


STATUS

approved



