calculate incremental inverse square values from $0$ to a value

Question

I would like to calculate $20$ inverse square values from $897–3773$, but am unsure how to really even begin.

I am attempting to set the voltage on a servo motor that moves a magnet away from a metal source, such that I have a constant difference between each setting ($20$ of them), and am fairly certain this means I need to move the magnet smaller increments at first, then grow larger.

For example I know the following values:

 x, y
-------
 1, 3773
 2, 3662
 5, 3336
10, 2677
15, 2064
19, 1293
20, 897

These are voltage reads, so not exact, but even plotting by hand it's clear it's not linear.

Any advice would be much appreciated. If anything, how to better think about the problem.

Answer 1

If I read your question correctly you want to interpolate $19$ values (so $20$ steps) between $x=897$ and $x=3773$ on the $x$ axis so that the function $y=1/x^2$ has constant increments.

To do that, divide the interval between $1/897^2$ and $1/3773^2$ into equal intervals. For each division point $p$ the value of $x$ will be $1/\sqrt{p}$.