A133108 - OEIS

%I #19 Feb 02 2022 10:36:43

%S 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,

%T 21,22,7,23,24,8,25,26,27,28,9,29,30,10,31,32,3,33,34,11,35,36,12,37,

%U 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

%N Representation of a dense para-sequence.

%C (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.

%D C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

%H Clark Kimberling, <a href="/A133108/b133108.txt">Table of n, a(n) for n = 1..10000</a>

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Kimberling/kimber16.html"> Self-Containing Sequences, Selection Functions, and Parasequences</a>, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

%e 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.

%t Flatten@NestList[Riffle[Range[Length[#] + 1, 3 Length[#] + 2], #, 3] &, {1, 2}, 3] (* _Birkas Gyorgy_, Mar 11 2011 *)

%Y Cf. A131987.

%K nonn

%O 1,2

%A _Clark Kimberling_, Sep 12 2007