

A232890


Denominator of nth term of sequence (or tree) S of all rational numbers generated by these rules: 0 is in S; if x is in S then x + 1 is in S, and if x + 1 is nonzero, then 1/(x + 1) is in S; duplicates are deleted as they occur.


2



1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 2, 3, 1, 5, 4, 3, 3, 5, 2, 5, 3, 1, 1, 6, 5, 4, 4, 7, 3, 8, 5, 2, 2, 7, 5, 3, 3, 4, 1, 7, 6, 5, 5, 9, 4, 11, 7, 3, 3, 11, 8, 5, 5, 7, 2, 9, 7, 5, 5, 8, 3, 7, 4, 1, 1, 8, 7, 6, 6, 11, 5, 14, 9, 4, 4, 15, 11, 7, 7, 10, 3
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OFFSET

1,5


COMMENTS

Let S be the sequence (or tree) of numbers generated by these rules: 0 is in S; if x is in S then x + 1 is in S, and if x + 1 is nonzero, then 1/(x + 1) is in S. Deleting duplicates as they occur, the generations of S are given by g(1) = (0), g(2) = (1,1), g(3) = (2,1/2), g(4) = (3, 1/3, 1/2, 2), ... Concatenating gives 0, 1, 1, 2, 1/2, 3, 1/3, 1/2, 2, 4, 1/4, ...
Conjectures: If b/c is a positive rational number, the position of n + b/c for n >= 0 forms a linear recurrence sequence with signature (1,1), and the position of n  b/c forms a linear recurrence sequence with signature (4, 4, 1). For n>=1, the numbers (1 + 1/n) are terminal nodes in the tree, and their positions are linearly recurrent with signature (2,0,1). For n >=3, the nth generation g(n) consists of F(n1) positive numbers and F(n1) negative numbers, where F = A000045, the Fibonacci numbers.


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000


EXAMPLE

To generate S, the number 0 begets (1,1), whence 1 begets 2 and 1/2, whereas 1 begets 0 and 1/2, both of which are (deleted )duplicates, so that g(3) = (2, 1/2). The resulting concatenation of all the generations g(n) begins with 0, 1, 1, 2, 1/2, 3, 1/3, 1/2, 2, 4, 1/4, so that A232890 begins with 1,1,1,1,2,1,3,2,1,1,4.


MATHEMATICA

Off[Power::infy]; x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, 1/(x + 1)} /. ComplexInfinity > 0]]], {8}]; x
On[Power::infy]; Denominator[x] (* Peter J. C. Moses, Nov 29 2013 *)


CROSSREFS

Cf. A232559, A232868, A000045.
Sequence in context: A266640 A065120 A176206 * A300441 A295665 A103484
Adjacent sequences: A232887 A232888 A232889 * A232891 A232892 A232893


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Dec 02 2013


STATUS

approved



