yeah thats a good example and it shows weird the number 0 is compared to the positive integers. it seems like a lot of the time things are first "defined" for the positive integers and then afterwards the definition is extended to 0 in a "consistent way". for example, the idea of taking exponents a^n^ makes sense when n is a positive integer, but its not immediately clear how to define a^0^. so, we do some digging and see that a^m+n^ = a^m^a^n^ when m and n are positive integers. this observation makes defining a^0^=1 "consistent" with the definition on positive integers, since it makes a^m+n^ = a^m^a^n^ true when n=0.

i think this sort of thing makes mathematicians think of 0 as a weird index and its why they tend to prefer starting at 1, and then making 0 the index for the "weird" term when it's included (like the displacement vector in affine space or the constant term in a taylor series).