Flat Maze sequence

[Eric Angelini]

Hello math-fun and SeqFan,

I don't know if this is old hat -- if yes please ignore.

Rules for a new arrangement of the natural integers :

0) At the beginning there was a (semi)-infinite row of
   empty labels;
1) Then came the Pencil who wrote "1" on the first (most-
   left) one;
2) "1" meant: "write now whatever integer you want (but
   not used so far) on the label to the right of me at a
   distance of "1" step;
   [and, more generally, an odd integers means: "go 2k+1
   labels to the right and write smthg new on it -- the
   label must be blank, though". An even integers means:
   "go 2k labels to the left and write...".]

Question: is there an algorithm to rearrange all the na-
tural integers with those two simple rules (leaving no
empty labels behind) ?

I have started this possible sequence (a dot means "still
empty label"; the next label to be named has an "x"):

        1   3   5   7   2  11  15   4  17  13
        9   6  21  23  25  27  10   .   .   8
        .  12  14   .  17  19   .   .   .   .
        .   .   .  20   .   .  22   .   .  24
        .   x  18   .  32   .   .   .   .   .

This sequence looks like an "flat maze" -- thus the name.
The above has been done by hand -- this explains why the
rows are 10 integers long: labels are easier to reach be-
cause a vertical step (downwards) adds 10 to the count.

Best,
É.

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