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%I #13 Jan 18 2022 03:37:34
%S 1,2,-1,3,-1,0,4,-1,1,5,-1,2,3,-2,6,-1,3,5,-3,5,-2,7,-1,4,7,-4,8,-3,7,
%T -2,1,8,-1,5,9,-5,11,-4,11,-3,2,9,-2,3,4,-3,9,-1,6,11,-6,14,-5,15,-4,
%U 3,14,-3,5,7,-5,11,-2,5,8,-5,7,-3,10,-1,7,13,-7
%N Numerators of rational numbers as generated by the rules: 1 is in S, and if nonzero x is in S, then x+1 and -1/x are in S. (See Comments.)
%C Let S be the set of numbers defined by these rules: 1 is in S, and if nonzero x is in S, then x + 1 and -1/x are in S. Then S is the set of all rational numbers, produced in generations as follows: g(1) = (1), g(2) = (2, -1), g(3) = (3, -1/2, 0), g(4) = (4, -1/3, 1/2), ... For n > 4, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements. Let S' denote the sequence formed by concatenating the generations.
%C A226130: Denominators of terms of S'
%C A226131: Numerators of terms of S'
%C A226136: Positions of positive integers in S'
%C A226137: Positions of integers in S'
%H Clark Kimberling, <a href="/A226131/b226131.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Fo#fraction_trees">Index entries for fraction trees</a>
%e The denominators and numerators are read from the rationals in S':
%e 1/1, 2/1, -1/1, 3/1, -1/2, 0/1, 4/1, -1/3, 1/2, ...
%t g[1] := {1}; z = 20; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]]&[DeleteCases[g[n - 1], 0]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; Flatten[Map[g, Range[7]]] (* ordered rationals *)
%t Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
%t f = Flatten[Map[g, Range[z]]];
%t Take[Denominator[f], 100] (* A226130 *)
%t Take[Numerator[f], 100] (* A226131 *)
%t p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
%t p2 = Flatten[Table[Position[f, -n], {n, 0, z}]];
%t Union[p1, p2] (* A226137 *) (* _Peter J. C. Moses_, May 26 2013 *)
%Y Cf. A226080 (rabbit ordering of positive rationals).
%K sign,frac
%O 1,2
%A _Clark Kimberling_, May 28 2013
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