cp's OEIS Frontend

Decree that (row 1) = (1) and (row 2) = (3,2). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 3/x for each x in row n-1, and duplicates are rejected as they occur. Then a(n) = (number of numbers in row n); it appears that this sequence is not linearly recurrent.

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x+1 and 1/x are in S. Then S is the set of positive rational numbers, which arise in generations as follows: g(1) = (1/1), g(2) = (1+1) = (2), g(3) = (2+1, 1/2) = (3/1, 1/2), g(4) = (4/1, 1/3, 3/2), ... . Once g(n-1) = (g(1), ..., g(z)) is defined, g(n) is formed from the vector (g(1) + 1, 1/g(1), g(2) + 1, 1/g(2), ..., g(z) + 1, 1/g(z)) by deleting all elements that are in a previous generation. A226080 is the sequence of denominators formed by concatenating the generations g(1), g(2), g(3), ... . It is easy to prove the following:

(1) Every positive rational is in S.

(2) The number of terms in g(n) is the n-th Fibonacci number, F(n) = A000045(n).

(3) For n > 2, g(n) consists of F(n-2) numbers < 1 and F(n-1) numbers > 1, hence the name "rabbit ordering" since the n-th generation has F(n-2) reproducing pairs and F(n-1) non-reproducing pairs, as in the classical rabbit-reproduction introduction to Fibonacci numbers.

(4) The positions of integers in S are the Fibonacci numbers.

(5) The positions of 1/2, 3/2, 5/2, ..., are Lucas numbers (A000032).

(6) Continuing from (4) and (5), suppose that n > 0 and 0 < r < n, where gcd(n,r) = 1. The positions in A226080 of the numbers congruent to r mod n comprise a row of the Wythoff array, W = A035513. The correspondence is sampled here:

row 1 of W: positions of n+1 for n>=0,

row 2 of W: positions of n+1/2,

row 3 of W: positions of n+1/3,

row 4 of W: positions of n+1/4,

row 5 of W: positions of n+2/3,

row 6 of W: positions of n+1/5,

row 7 of W: positions of n+3/4.

(7) If the numbers <=1 in S are replaced by 1 and those >1 by 0, the resulting sequence is the infinite Fibonacci word A003849 (except for the 0-offset first term).

(8) The numbers <=1 in S occupy positions -1 + A001950, where A001950 is the upper Wythoff sequence; those > 1 occupy positions given by -1 + A000201, where A000201 is the lower Wythoff sequence.

(9) The rules (1 is in S, and if x is in S, then 1/x and 1/(x+1) are in S) also generate all the positive rationals.

A variant which extends this idea to an ordering of all rationals is described in A226130. - M. F. Hasler, Jun 03 2013

The updown and downup zigzag limits are (-1 + sqrt(5))/2 and (1 + sqrt(5))/2; see A020651. - Clark Kimberling, Nov 10 2013

From Clark Kimberling, Jun 19 2014: (Start)

Following is a guide to related trees and sequences; for example, the tree A226080 is represented by (1, x+1, 1/x), meaning that 1 is in S, and if x is in S, then x+1 and 1/x are in S (except for x = 0).

All the positive integers:

A243571, A243572, A232559 (1, x+1, 2x)

A232561, A242365, A243572 (1, x+1, 3x)

A243573 (1, x+1, 4x)

All the integers:

A243610 (1, 2x, 1-x)

A232723, A242364

All the positive rationals:

A226080, A226081, A242359, A242360 (1, x+1, 1/x)

A243848, A243849, A243850 (1, x+1, 2/x)

A243851, A243852, A243853 (1, x+1, 3/x)

A243854, A243855, A243856 (1, x+1, 4/x)

A243574, A242308 (1, 1/x, 1/(x+1))

A241837, A243575 ({1,2,3}, x+4, 12/x)

A242361, A242363 (1, 1 + 1/x, 1/x)

A243613, A243614 (0, x+1, x/(x+1))

All the rationals:

A243611, A243612 (0, x+1, -1/(x+1))

A226130, A226131 (1, x+1, -1/x)

A243712, A243713 ({1,2,3}, x+1, 1/(x+1))

A243730, A243731 ({1,2,3,4}, x+1, 1/(x+1))

A243732, A243733 ({1,2,3,4,5}, x+1, 1/(x+1))

A243714, A243715

A243925, A243926, A243927 (1, x+1, -2/x)

A243928, A243929, A243930 (1, x+1, -3/x)

All the Gaussian integers:

A243924 (1, x+1, i*x)

All the Gaussian rational numbers:

A233694, A233695, A233696 (1, x+1, i*x, 1/x).

(End)

Decree that (row 1) = (1). For n >= 2, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 4/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.