Problem 155

Problem 155

Counting Capacitor Circuits

An 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 60μF 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 C1, C2 etc in parallel, the total capacitance is CT = C1 + C2 +…,
whereas when connecting them in series, the overall capacitance is given by:$\frac{1}{C_T}=\frac{1}{C_1}+\frac{1}{C_2}+…$.




我们用D(n)表示当我们使用最多n个相同的电容并重复上述简单过程后,我们能够得到的总电容的种数,我们有:D(1)=1, D(2)=3, D(3)=7 …


提醒:当电容C1,C2等等并联时,总电容为CT = C1 + C2 +…,