

A226136


Positions of the positive integers in the ordering 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).


5



1, 2, 4, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982
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OFFSET

1,2


COMMENTS

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(n1) = (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.
A226130: Denominators of terms of S'
A226131: Numerators of terms of S'
A226136: Positions of positive integers in S'
A226137: Positions of integers in S'


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..35


FORMULA

Conjecture: a(n) = a(n1)+a(n3) for n>6. G.f.: x*(x+1) * (x^2+1)^2 / (x^3+x1).  Colin Barker, Jul 03 2013


EXAMPLE

S' = (1/1, 2/1, 1/1, 3/1, 1/2, 0/1, 4/1, 1/3, 1/2, ...), with positive integers appearing in positions 1,2,4,7,...


MATHEMATICA

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 *)
Map[g, Range[z]]; Table[Length[g[i]], {i, 1, z}] (* cf A003410 *)
f = Flatten[Map[g, Range[z]]];
Take[Denominator[f], 100] (* A226130 *)
Take[Numerator[f], 100] (* A226131 *)
p1 = Flatten[Table[Position[f, n], {n, 1, z}]] (* A226136 *)
p2 = Flatten[Table[Position[f, n], {n, 0, z}]];
Union[p1, p2] (* A226137 *) (* Peter J. C. Moses, May 26 2013 *)


CROSSREFS

Cf. A226080 (rabbit ordering of positive rationals).
Sequence in context: A179385 A024668 A188951 * A176099 A160790 A173726
Adjacent sequences: A226133 A226134 A226135 * A226137 A226138 A226139


KEYWORD

nonn


AUTHOR

Clark Kimberling, May 28 2013


STATUS

approved



