A133108 Representation of a dense para-sequence.

1, 2, 3, 4, 1, 5, 6, 2, 7, 8, 9, 10, 3, 11, 12, 4, 13, 14, 1, 15, 16, 5, 17, 18, 6, 19, 20, 2, 21, 22, 7, 23, 24, 8, 25, 26, 27, 28, 9, 29, 30, 10, 31, 32, 3, 33, 34, 11, 35, 36, 12, 37, 38, 4, 39, 40, 13, 41, 42, 14, 43, 44, 1, 45, 46, 15, 47, 48, 16, 49, 50, 5, 51, 52, 17, 53, 54, 18

Offset: 1

Author

Clark Kimberling, Sep 12 2007

Keywords

• nonn

Comments

(1) A fractal sequence. (2) The para-sequence may be regarded as a sort of "limit" of the concatenated segments. The para-sequence (itself not a sequence) is dense in the sense that every pair of terms i and j are separated by another term (and hence separated by infinitely many terms). (3) The para-sequence accounts for positions of triadic rational numbers in the following way: 1/3 < 2/3 matches the segment 1,2; 1/9 < 2/9 < 1/3 < 4/9 < 5/9 < 2/3 < 7/9 < 8/9 matches the segment 3,4,1,5,6,2,7,8, etc.

Examples

			The first segment is 1,2; the 2nd is 3,4,1,5,6,2,7,8; the 4th begins with 27,28,9 and ends with 26,79,80.
		

References

• Clark Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Crossrefs

Cf. A131987.

Programs

Mathematica

Flatten@NestList[Riffle[Range[Length[#] + 1, 3 Length[#] + 2], #, 3] &, {1, 2}, 3] (* Birkas Gyorgy, Mar 11 2011 *)