A132223 A dense infinitive sequence.

1, 2, 1, 4, 2, 3, 1, 8, 4, 7, 2, 6, 3, 5, 1, 16, 8, 15, 4, 14, 7, 13, 2, 12, 6, 11, 3, 10, 5, 9, 1, 32, 16, 31, 8, 30, 15, 29, 4, 28, 14, 27, 7, 26, 13, 25, 2, 24, 12, 23, 6, 22, 11, 21, 3, 20, 10, 19, 5, 18, 9, 17

Offset

1,2

Comments

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

References

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

Links

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.

Example

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.

Maple

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A132224.

Sequence in context: A341392 A351886 A323901 * A224712 A363301 A135941

Adjacent sequences: A132220 A132221 A132222 * A132224 A132225 A132226

Keyword

nonn

Author

Clark Kimberling, Aug 14 2007

Status

approved