2020 HMMT November General Round Problem 8

A bar of chocolate is made of 10 distinguishable triangles as shown below:

How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces?

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2020 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.