This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A232890 #5 Dec 04 2016 19:46:33
%S A232890 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,
%T A232890 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,
%U A232890 1,8,7,6,6,11,5,14,9,4,4,15,11,7,7,10,3
%N 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.
%C A232890 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, ...
%C A232890 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.
%H A232890 Clark Kimberling, <a href="/A232890/b232890.txt">Table of n, a(n) for n = 1..1000</a>
%e A232890 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.
%t A232890 Off[Power::infy]; x = {0}; Do[x = DeleteDuplicates[Flatten[Transpose[{x, x + 1, -1/(x + 1)} /. ComplexInfinity -> 0]]], {8}]; x
%t A232890 On[Power::infy]; Denominator[x] (* _Peter J. C. Moses_, Nov 29 2013 *)
%Y A232890 Cf. A232559, A232868, A000045.
%K A232890 nonn,easy
%O A232890 1,5
%A A232890 _Clark Kimberling_, Dec 02 2013