Kimberling Sequence

A sequence generated by beginning with the Positive integers, then iteratively applying the following algorithm:

1. In iteration $i$, discard the $i$th element,

2. Alternately write the $i+k$ and $i-k$th elements until $k=i$,

3. Write the remaining elements in order.

The first few iterations are therefore

$$\matrix{ \lower4pt\vbox{\hrule\hbox{\vrule\kern3pt\vbox{\hbo... ...n3pt}\kern3pt\vrule}\hrule } & 6 & 11 & 12 & 13 & 14 & 15\cr}.$$

The diagonal elements form the sequence 1, 3, 5, 4, 10, 7, 15, ... (Sloane's A007063).

References

Guy, R. K. ``The Kimberling Shuffle.'' §E35 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 235-236, 1994.

Kimberling, C. ``Problem 1615.'' Crux Math. 17, 44, 1991.

Sloane, N. J. A. Sequence A007063/M2387 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.