One can express the finite differences in the previous post with the values of y' to obtain a recursion formula that simplifies the calculation of the table.
Again one does an initial sample calculation using the recursion formula and then selects it and drags down to repeat the calculation for successive entries.
Computer programmers use recursion formulas like this to reduce the number of calculations needed to do a given computation. One does not need to know the coefficients of the cubic equation to calculate the entire curve; the first four points are sufficient.
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