This array is formed by placing integers such that each is the smallest such that the rectangle with it and the top left hand corner as opposite corners does not contain the same integer twice in any row or column.
$\begin{array}\\ 1&2&3&4&5&6&7&8&9 \ \ ...\\ 2&1&4&3&6&5&8&7&10\\ 3&4&1&2&7&8&5&6&11\\ 4&3&2&1&8&7&6&5&12\\ 5&6&7&8&1&2&3&4&13\\ \vdots \end{array}$
The diagonal sums form the sequence $p_1(n)=1,4,7,16,17,28,39,64,57,68,79,112,121,156,191,256,225,228,231...$
Since the array is symmetric and each $2^n$ by $2^n$ submatrix and every $2^{nd}$ row is made by swapping pairs in the row above and every $4^{th}$ row is made by swapping pairs of pairs in the row $2$ above etc. $2^{k}|p_1(n) \ \text{iff} \ 2^{\lceil\frac{k}2\rceil}|n$.
This array is the $1^{st}$ plane of a latin cube where the $2^{nd}$ plane would be
$\begin{array}\\ 2&1&4&3&6&5&8&7&10 \ \dots\\ 1&2&3&4&5&6&7&8&9\\ 4&3&2&1&8&7&6&5&16\\ 3&4&1&2&7&8&5&6&11\\ \vdots \end{array}$
With diagonal sums $p_2(n)=2,2,10,12,22,22,46,56,66,58,98,108...$
$2|p_2(n) \forall n \\ 4|p_2(n) \text{ iff } 4|n \\ 8|p_2(n) \text{ iff } 8|n$
Plane $3$:
$ \begin{array}\\ 3&4&1&2&7 \ \dots\\ 4&3&2&1&8\\ \vdots \end{array}$
$p_3(n)=3,8,5,8,19,40,37,48,59,88,77,88...\\ p_4(n)=4,6,8,4,24,34,44,40,68...\\ p_5(n)=5,12,19,32,21,20,19,32,45...$
When does $2^k|p_i(n)$ in general?
The sums of the elements on diagonal planes make the sequence $s(n)=1,6,12,38,45,72... \text{ where } s(n)=\sum_{i=1}^n p_i(n-i+1)$. Is every third number in this sequence a multiple of $4$?
What happens if this is the $1^{st}$ cube in an infinite latin tessarect?