WOLFRAM|DEMONSTRATIONS PROJECT

Successive Differences of Sequences

​
sequence
powers of 15
length of sequence
20
modulus
211
color
Let
s
be a sequence of numbers
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,…
. In this Demonstration the sequences are named integer sequences.
The sequence of differences of
s
is a new sequence
Δ(s)=
a
2
-
a
1
,
a
3
-
a
2
,
a
4
-
a
3
,
a
5
-
a
4
,…
.
For example, the sequence of differences of the sequence
1,3,9,27,81,243,…
is
2,6,18,54,162,486,…
. The second differences are the differences of the differences, in this case
4,12,36,108,324,972,…
.
In the graphic, numbers are coded by color. Each row consists of the differences of the row above it, shifted over by one each time. In symbols, the first row is the sequence
s
; the second row is the sequence
Δ(s)
; the third row is
Δ(Δ(s))=
2
Δ
(s)
; and so on.
Taking differences is the first thing to try when investigating a sequence.