Convex numbers

Take the number 134431001 as an example. We can make pairs using the 2 digits following one another: [(1, 3), (3, 4), (4, 4), (4, 3), (3, 1), (1, 0), (0, 0), (0, 1)] If we assume that these are the lattice points, we can draw line segments in the same order and we can have a polygon. In general for a k-digit number like d₁d₂...d_k we can have a (k-1)-gon: [(d₁,d₂), (d₂,d₃), ..., (d_k-1,d_k)] If the drawn polygon is a convex one then we will call this kind of number as convex number. How many positive convex numbers are there? Answer format: count,sum [My timing: < 1m] Notes: - Any two neighbouring line segments cannot be collinear. - All the generated pairs must be distinct. - Some polygons can be generated by more than one number. For example: 112321, 123211, 232112, 321123, 211232 Since the numbers are different, they are all counted.