My experience is that 0-based offsets (and use of < or even != for upper bounds) mean that I should almost never have to write something like idx - 1 or idx + 1.

I came to 0-based offsets later in my career, having started with Matlab. So I have some real experience with 1-based offsets. Experience that was 'untainted' by being used to a different option. I much prefer 0-based.

Especially because I now sort-off have a linter rule in my head 'if I am writing i - 1 then I am making a mistake or doing something the wrong way'. Which has been quite successful.

I have a similar experience. Matlab was my first experience with computational work. Now I use Python and 0-based is still more natural.

Not for polynomial coefficient indices :-)

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.