Or Fourier coefficients :-) Or pretty much anything where the index/subscript is related to the math itself.

Math textbooks and papers tend to use 1-based subscripts when it doesn't matter. It's hard to come up with examples where starting at 1 facilitates the actual math.

Just for consistency: a_n is the coefficient of x^n, so the constant term ends up being a_0. Based on my experience, numbering starts from one (like (x_1, x_2, x_3) as point of R^3) and off-sets from zero, e.g., when dealing with discrete time, t_0 is the first.