Padovan Triangular Spiral

This is a generalized version of the Padovan triangle spiral. What makes the pattern spiral geometrically? In other words, how do you know where to draw the next triangle? When the initial golden triangle is equilateral, there is an interesting pattern of edge lengths, known as the Padovan sequence. What is the pattern? Why does the spiral share it?
More GeoGebra at More on the Padovan spiral at Wikipedia: