[defrost]

Fence post and rail is "off by one", two fence posts are joined by one rail, N fence posts are joined by N-1 rails, and this polynomial order and defining coefficients discussion has that in common.

Two points define a line, a polynomial of degree 1. A polynomial with 2 coefficients, ax + b.

Three points give us a quadratic, a polynomial of degree 2 with three coefficients, ax^2 + bx + c.

N points gives us a polynomial of degree N-1 with N coefficients.

Indexing coefficients by their associated power of X seems natural to some.

A(N-1).X^(N-1) + ... A(1).X^1 + A(0).X^0 (where X^0 == 1)

are the N indexed coefficients of a generic polynomial of order N-1.