## Counting Capacitor Circuits

### Problem 155

Published on Saturday, 19th May 2007, 02:00 am; Solved by 2035An electric circuit uses exclusively identical capacitors of the same value C.

The capacitors can be connected in series or in parallel to form sub-units, which can then be connected in series or in parallel with other capacitors or other sub-units to form larger sub-units, and so on up to a final circuit.

Using this simple procedure and up to `n` identical capacitors, we can make circuits having a range of different total capacitances. For example, using up to `n`=3 capacitors of 60 F each, we can obtain the following 7 distinct total capacitance values:

If we denote by `D`(`n`) the number of distinct total capacitance values we can obtain when using up to `n` equal-valued capacitors and the simple procedure described above, we have: `D`(1)=1, `D`(2)=3, `D`(3)=7 ...

Find `D`(18).

*Reminder :* When connecting capacitors C_{1}, C_{2} etc in parallel, the total capacitance is C_{T} = C_{1} + C_{2} +...,

whereas when connecting them in series, the overall capacitance is given by: