Find the Kimberling Sequence

How to find the Kimberling Sequence by using Mathematica?

each row is obtained from the previous by boxing (and expelling) the main diagonal element, and then reading the first number after the box, the first before the box, the second after the box, the second before the box and so on until all the initial numbers are read off..

Kimberling Sequence is a diagonal elements form the sequence $$=1,3,5,4,10...$$

I have no idea how to define this sequence in Mathematica.

Thank you in advance to any one who may be able to give me some ideas

(There is a unsolved problem about this sequence, $$\textbf{ "Is every positive integer a term of this sequence?"}$$ )

• There's code here; but I suspect there are more efficient ways to do this. – J. M.'s discontentment May 31 '16 at 14:07