In AutoCAD this would be a simple enough operation with the Array command. Unfortunately I'm not aware of any similar tool or command in ArcGIS (perhaps someone else is). It can be done with scripting, and there is at least one question here that covers that.
However I can offer a simple if slightly time consuming method. Note I'm going to make a regular grid assumption about your arrangement. You may want to provide a graphic example if what you want is different, say for them to 'lean' against each other in a row as fallen dominoes would (see second image). 'Efficiently as possible' is a little open to interpretation.
- Start by running the Create Fishnet tool. This will create a
reference grid for your ellipses. You'll specify an origin for the
grid, and then a coordinate that in combination with the origin
defines the y axis. Using that coordinate you can create your slight
angle. Your cell width will be half the major axis and height half
the minor axis of the desired elipse. Number of rows and columns
should be roughly twice the number of desired shapes in either
direction. You won't need label points, and lines or polygons is up
to you and will just affect method of snap in later steps.
- With your fishnet created, now you can start editing the layer for
your elipses. Create Feature and pick the elipse construction tool,
then click on the origin of the net, then the first x intersection,
then the first xy intersection. This should get you your first
- To create the rest, you have options. You can keep drawing them one
by one or you can start copying. Using copy, you can exponentially
grow the array with each copy: Select the first, right click and
choose copy, right click and choose paste, then drag the duplicated
elipse (sitting on top of your original) and snap the center to the
second x intersection. Then select both, copy, drag and snap further
along the x axis. Then select all four, copy, drag and snap plus two
on the y axis. Now you have eight you can replicate, and so on.
If you're looking for a different pattern, you should be able to adjust the grid mathematically based on the desired relationship between ellipse centers. The below image is an example of the domino pattern I mentioned earlier, but by hand without a grid reference.