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A131967 Farey fractal sequence. 4
1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 6, 4, 3, 5, 7, 2, 1, 8, 6, 4, 9, 3, 10, 5, 7, 11, 2, 1, 12, 8, 6, 4, 9, 3, 10, 5, 7, 11, 13, 2, 1, 14, 12, 8, 6, 15, 4, 9, 16, 3, 17, 10, 5, 18, 7, 11, 13, 19, 2, 1, 20, 14, 12, 8, 6, 15, 4, 21, 9, 16, 3, 17, 10, 22, 5, 18, 7, 11, 13, 19, 23, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

As a fractal sequence, A131967 properly contains itself as a subsequence (infinitely many times).

Step 1: List the Farey fractions by order, like this:

order 1: 0/1 1/1

order 2: 0/1 1/2 1/1

order 3: 0/1 1/3 1/2 2/3 1/1, etc.

Step 2: Replace each a/b by its position when all the segments in Step 1 are concatenated and each distinct predecessor of a/b is counted just once, getting

1 2

1 3 2

1 4 3 5 2, etc.

Step 3: Concatenate the segments found in Step 2.

REFERENCES

C. Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

LINKS

Table of n, a(n) for n=1..83.

EXAMPLE

The Farey fractions of order 4 are

0 1/4 1/3 1/2 2/3 3/4 1, having position numbers

1 6 4 3 5 7 2, which is the fourth segment in the formation of A131967.

MATHEMATICA

Farey[n_] :=

  Select[Union@

    Flatten@Outer[Divide, Range[n + 1] - 1, Range[n]] , # <= 1 &];

newpos[n_] :=

  Module[{length = Total@Array[EulerPhi, n] + 1, f1 = Farey[n],

    f2 = Farey[n - 1], to},

   to = Complement[Range[length], Flatten[Position[f1, #] & /@ f2]];

   ReplacePart[Array[0 &, length],

    Inner[Rule, to, Range[length - Length[to] + 1, length], List]]];

a[n_] := Flatten@

   Table[Fold[

     ReplacePart[Array[newpos, i][[#2 + 1]],

       Inner[Rule,

        Flatten@Position[Array[newpos, i][[#2 + 1]], 0], #1, List]] &,

      Array[newpos, i][[1]], Range[i - 1]], {i, n}];

a[10] (* Birkas Gyorgy, Feb 21 2011 *)

CROSSREFS

Cf. A131968.

Sequence in context: A133404 A134627 A064881 * A329501 A300670 A137679

Adjacent sequences:  A131964 A131965 A131966 * A131968 A131969 A131970

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 02 2007

STATUS

approved

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Last modified June 19 20:27 EDT 2021. Contains 345145 sequences. (Running on oeis4.)