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A232890 Denominator of n-th 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. 1
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 (list; graph; refs; listen; history; text; internal format)
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 n-th generation g(n) consists of F(n-1) positive numbers and F(n-1) 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 * A103484 A016444 A280831

Adjacent sequences:  A232887 A232888 A232889 * A232891 A232892 A232893

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Dec 02 2013

STATUS

approved

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Last modified July 26 00:44 EDT 2017. Contains 289798 sequences.