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 A32224 a fractal sequence.

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

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.

Cf. A132224.

Sequence in context: A123755 A118291 A118290 this_sequence A135941 A036998 A121464

Adjacent sequences: A132220 A132221 A132222 this_sequence A132224 A132225 A132226

nonn

Clark Kimberling (ck6(AT)evansville.edu), Aug 14 2007