I can't tell the answer to your sequence, but I can tell some techniques to find a natural pattern in a finite sequence. First you should understand what it means to "differentiate" finite sequences. The idea I will use will be similar as "integrating the derivative" to find the function.
Define $x_n =$ the $n^{\text{th}}$ term of your sequence and let $\Delta x_n = x_{n+1} - x_n$. Then the sequence $\Delta x_n$ looks like this : $ 24 \, 27 \, 33 \, 45 \, 69 \, 96 \, 90 \, 90 \, 150 \, 180 \, 150 \, 210 \, 240 \, 330 \, 330 \,... $ In this case it doesn't help, but it usually does. You can repeat this process until you find a nice pattern (it might not work), and then compute your sequence's formula inductively using $x_{n+1} = x_n + \Delta x_n$.