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The commonly known mathematical formulas for calculating the mutual inductance of thin coaxial cylindrical coils (thin wall solenoids) in air can be modified and generalized so that they are also applicable for the relatively simple and versatile calculation of the mutual inductance of thin non-coaxial coils with parallel axes. This is a geometric arrangement whereby one coil is circular cylindrical and the other can be of any requisite shape (e.g., square, rectangular). This manner of calculation is demonstrated in this article using as an example two thin wall solenoids and also for a situation in which one coil is circular cylindrical and the second is rectangular. Detailed algorithms have been developed for both these situations, which can serve for the creation of a computer program.