Build a $5\times5$ (say) square using $\frac{1-\left(\frac{3}{2}\right)^n}{(2 m-1) 2^n-1}+1$, $$ \left( \begin{array}{ccccc} 0.5\hfill & 0.9\hfill & 0.944444 & 0.961538 & 0.970588 \\ 0.583333 & 0.886364 & 0.934211 & 0.953704 & 0.964286 \\ 0.660714 & 0.896739 & 0.939103 & 0.956818 & 0.966549 \\ 0.729167 & 0.913564 & 0.948576 & 0.963401 & 0.971591 \\ 0.787298 & 0.930592 & 0.95853\hfill & 0.970432 & 0.977025 \\ \end{array} \right). $$ Every row is ordered. Every column (from row 2 on) is ordered. The nw-se diagonal is ordered. I claim there are no duplicate numbers and that you can build as large a square as you want and everything still holds.
Is there a name for this square? Links to some resources would be nice.
Edit Rotate 90 degrees left and this becomes the pattern of the left square side of a triangular prism lattice representing a stack of Nicomachus' Triangles. The usual values would be integers as shown below for a $10\times10$ square.
$$\left(
\begin{array}{cccccccccc}
38 & 76 & 152 & 304 & 608 & 1216 & 2432 & 4864 & 9728 & 19456 \\
34 & 68 & 136 & 272 & 544 & 1088 & 2176 & 4352 & 8704 & 17408 \\
30 & 60 & 120 & 240 & 480 & 960 & 1920 & 3840 & 7680 & 15360 \\
26 & 52 & 104 & 208 & 416 & 832 & 1664 & 3328 & 6656 & 13312 \\
22 & 44 & 88 & 176 & 352 & 704 & 1408 & 2816 & 5632 & 11264 \\
18 & 36 & 72 & 144 & 288 & 576 & 1152 & 2304 & 4608 & 9216 \\
14 & 28 & 56 & 112 & 224 & 448 & 896 & 1792 & 3584 & 7168 \\
10 & 20 & 40 & 80 & 160 & 320 & 640 & 1280 & 2560 & 5120 \\
6 & 12 & 24 & 48 & 96 & 192 & 384 & 768 & 1536 & 3072 \\
2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024 \\
\end{array}
\right)
$$
The diagonal with non-decreasing ordered positive integers is sw-ne. Other orders are up, left to right, and nw-se.
I'm interested about what might be said about the pattern.