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--