A

technique for analysing

sequences, from a given

sequence f(n), a new

sequence
df(n) is formed which consists of the

difference of

successive terms of f(n).
This process is

iterated and if eventually the resulting

sequence is

constant,
the original

sequence can be reconstructed from the

finite differences evaluated
at some fixed n.

For instance the

perfect cubes have the following

finite difference table:

0 1 8
27 64 125
216 343 512
729 1000 . . .

1 7 19
37 61 91
127 169
217 271 . . .

6 12 18
24 30 36
42 48
54 . . .

6 6
6 6
6 6
6 6
. . .

The

cubes could be reconstructed from the numbers (0,1,6,6) using only

addition.

--back to

combinatorics--