What equation should I use to calculate inductance of wire coil?
The proper formula depends upon a) whether your coil has a core and b) whether your coil is long or short, fat or thin, etc.
The most efficient air-core coil in terms of copper is a Brooks coil. It has a square cross section, and an outside diameter twice its inside diameter. You can see an image of the coil here.
The formula for the inductance of a Brooks coil is:
$$L \approx 0.025491rN^2$$
where N is the number of turns,
r is the inside radius in cms,
and L is the inductance in uH
There are also formulae for single layer long coils, multi-layer short coils and flat &quot;pancake&quot; coils found at this page
This image from AllAboutCircuits, shows various coil cross sections, and their corresponding (approximate) inductance formulae.