Consider a sequence C = (C_{0}, C_{1}, C_{2}, …). Define:
A_{0,j} = C_{j},
A_{i,j} = A_{i,j+1} − A_{i,j}, and
R_{i} = A_{i,o}.
This might be pictured:
C_{0} | C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | ||||||||||
|| | || | || | || | || | || | ||||||||||
R_{0} | = | A_{0,0} | A_{0,1} | A_{0,2} | A_{0,3} | A_{0,4} | A_{0,5} | … | |||||||
R_{1} | = | A_{1,0} | A_{1,1} | A_{1,2} | A_{1,3} | A_{1,4} | … | ||||||||
R_{2} | = | A_{2,0} | A_{2,1} | A_{2,2} | A_{2,3} | … | |||||||||
R_{3} | = | A_{3,0} | A_{3,1} | A_{3,2} | … |
R = (R_{0}, R_{1}, R_{2}, …) is another sequence. We might call C the sequence generated by R and R the generator of C.
It turns out that
;
.
The first of these formulae is particularly handy if the the sequence R is finite (or rather, is eventually zero):
.
Given some unknown sequence, this formula may be used to find empirically the power-series closed form, if it has one.
For example, consider the tetrahedral numbers C = (0, 1, 4, 10, 20, 35, …). We find that R = (0, 1, 2, 1).
Therefore .
This is not a proof of course, but it can be a useful experimental tool. It is also useful for manipulating sum sequences and difference sequences.
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