What is the pattern for resonant frequencies of a tube closed at BOTH ends?
The cases for the string, the open tube and the tube closed at one ends are well known.
The rule for tubes is that you need a node on each closed end of the tube. This means that the tube closed on both ends is EXACTLY equivalent to a string. The equation that applies will thus be $f_n=n(v/2L)$, where $n=1, 2, 3, ...$.