1,2

The sequence is dense in the sense that any two neighboring terms in a segment are separated in all succeeding segments. Thus in the limiting para-sequence, each pair of positive integers are separated by infinitely many positive integers.

A sequence is an infinitive sequence if and only if it is a sequence that contains every positive integer and also contains itself as a proper subsequence.

See A132224 for the normalization of A132223, making A132224 a fractal sequence.

Imagine the following magic trick: A magician places a stack of 8 cards face-down on a table. Then he transfers the top card to the bottom of the stack and deals the new top card face-up on the table. He repeats the procedure until all the cards are dealt. And —- abracadabra! —- the cards are in decreasing order. To perform the trick, the magician needs to arrange the cards in advance according to step three described in this sequence: 1, 8, 4, 7, 2, 6, 3, 5. Step n allows the trick to be performed with 2^n cards. - Tanya Khovanova, Feb 19 2022

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

T. D. Noe, Table of n, a(n) for n = 1..8190 (12 segments)

Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Start with 1,2. Separate them by 3,4, like this: 1,4,2,3. Then separate those by 5,6,7,8 like this: 1,8,4,7,2,6,3,5. Continue the process. Regard 1,2 and 1,4,2,3 and 1,8,4,7,2,6,3,5 as successive segments, so that the n-th segment has 2^n terms.

The next segment after 1,8,4,7,2,6,3,5, formed by separating those by 9,10,11,12,13,14,15,16, is 1,16,8,15,4,14,7,13,2,12,6,11,3,10,5,9.

A132223 := proc(n, k)

option remember ;

if n = 1 then

return k;

else

if type(k, 'odd') then

return procname(n-1, (k+1)/2) ;

else

return 2^n-k/2+1 ;

end if;

end if;

end proc:

seq(seq( A132223(n, k), k=1..2^n), n=1..8) ; # R. J. Mathar, May 08 2016

Flatten@FoldList[Riffle[#1, Range[2^#2, 2^(#2 - 1) + 1, -1]] &, {1, 2}, Range[2, 5]] (* Birkas Gyorgy, Apr 20 2011 *)

(Haskell)

import Data.List (inits)

a132223 n = a132223_list !! (n-1)

a132223_list = f 1 [1] where

f k xs = ys ++ f (2 * k) ys where

ys = concat $ transpose [xs, reverse $ take k [k+1 ..]]

-- Reinhard Zumkeller, Apr 14 2014

Cf. A132224.

Sequence in context: A341392 A351886 A323901 * A224712 A390938 A363301

Adjacent sequences: A132220 A132221 A132222 * A132224 A132225 A132226

Clark Kimberling, Aug 14 2007

approved