A trip down Mathematical lane

My friend and I were bored when we found this Signpost 8 Textbook. We opened a page, and the question was:

A level 1 card tower takes 2 cards to make
/\

A level 2 card tower takes 7 cards to make
  /_\
/ \/ \

A level 3 card tower takes 15 cards to make
     /_\
  /_\/_\
/ \ / \ / \
How many cards does a 10 tower card require to make?

-------------------------------

After trying to squeeze out every single drop of our high school maths knowledge, we eventually worked it out. Eventually, meaning like 20 minutes later. But we didn't actually work it out. We just counted it. Booo.....

We put the book away, and said we wouldn't do anymore.

But, somehow, I was really disappointed at myself. I couldn't bare the fact that I couldn't work it out. We worked out 10 because we counted, but what if the question asked for a level 100, or 10,000? Maybe it was my anal attitude, but I then decided that I HAD to work out the formula. My friend said it wasn't worth it. He was right.

But I won, and we spent our time trying to work it out.

..... eventually and happily, and an hour later, I finally worked it out. It's a bit hard to describe how it was done. I pretty much broke the puzzle up into two sections.
I wrote down out the first 4 required cards for the one's that weren't horizontal (being 2, 6, 12, 20), and through trial and error, working out the formula.
Then I wrote down the pattern for the cards that were horizontal (0, 1, 3, 6), and did the same.

Combining those two formulas gave me the overall formula.

Of course, this process would have taken 10 seconds by someone who remembers the formulas.

But still, I think, that has to be the greatest achievement for me last week.

Proud to be asian! (or not, lol)