|
| |
|
|
A226247
|
|
Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. (See Comments.)
|
|
9
|
|
|
|
1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 2, 3, 1, 6, 5, 4, 3, 3, 5, 2, 5, 3, 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1, 1, 8, 7, 6, 5, 5, 9, 4, 11, 7, 3, 11, 8, 5, 2, 2, 9, 7, 5, 3, 3, 4, 1, 9, 8, 7, 6, 6, 11, 5, 14, 9, 4, 15, 11, 7, 3, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,6
|
|
|
COMMENTS
|
Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. Then S is the set of all rational numbers, produced in generations as follows:
g(1) = (0), g(2) = (1), g(3) = (2, -1), g(4) = (3, -1/2), g(5) = (4, -1/3, 1/2), ... For n > 2, 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.
A226247: Denominators of terms of S''
A226248: Numerators of terms of S''
A226249: Positions of nonnegative numbers in S''
A226250: Positions of positive numbers in S''
A closely related sequence S' (for which the rules of generation are shorter but the resulting sequence is slightly less natural) is discussed at A226130. For both S' and S'', the number of numbers in g(n) is given by A097333.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The denominators and numerators are read from S'':
0/1, 1/1, 2/1, -1/1, 3, -1/2, 4/1, -1/3, 1/2, 5, -1/4, 2/3, 3/2, -2, ...
Table begins:
n |
--+-----------------------------------------------
1 | 1;
2 | 1, 1;
3 | 1, 2;
4 | 1, 3, 2;
5 | 1, 4, 3, 2, 1;
6 | 1, 5, 4, 3, 2, 2, 3;
7 | 1, 6, 5, 4, 3, 3, 5, 2, 5, 3;
8 | 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1;
|
|
|
MATHEMATICA
|
Clear[g]; z = 12; g[1] := {0}; g[2] := {1}; g[n_] := g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]] &[g[n - 1]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; f = Flatten[Map[g, Range[z]]]; Take[Denominator[f], 100] (*A226247*)
t = Take[Numerator[f], 100] (*A226248*)
s[n_] := If[t[[n]] > 0, 1, 0]; u = Table[s[n], {n, 1, Length[t]}]
Flatten[Position[u, 1]] (*A226249*)
|
|
|
PROG
|
(Python)
from fractions import Fraction
from itertools import count, islice
def agen():
rats = [Fraction(0, 1)]
seen = {Fraction(0, 1)}
for n in count(1):
yield from [r.denominator for r in rats]
newrats = []
for r in rats:
f = 1+r
if f not in seen:
newrats.append(1+r)
seen.add(f)
if r != 0:
g = -1/r
if g not in seen:
newrats.append(-1/r)
seen.add(g)
rats = newrats
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
