# [seqfan] The first differences of S are the odd terms of S (new)

Eric Angelini Eric.Angelini at kntv.be
Mon Dec 6 17:51:35 CET 2010

```Hello SeqFans,
(in full color here:

We start alternating even numbers and "holes" like this:
S = 2 . 4 . 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
We fill the first hole with '1':
S = 2 1 4 . 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
We compute the first differences, D:
S = 2 1 4 . 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
D =  1 3
We duplicate this last '3' in the first free hole of S:
S = 2 1 4 3 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
D =  1 3
We compute the next differences, D:
S = 2 1 4 3 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
D =  1 3 1 3
We duplicate these results accordingly in S:
S = 2 1 4 3 6 1 8 3 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
D =  1 3 1 3
We extend D as before:
S = 2 1 4 3 6 1 8 3 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
D =  1 3 1 3 5 7 5 7
Duplication in S of the new terms of D:
S = 2 1 4 3 6 1 8 3 10 5 12 7 14 5 16 7 18 . 20 . 22 .....
D =  1 3 1 3 5 7 5 7
Etc.
S = 2 1 4 3 6 1 8 3 10 5 12 7 14 5 16 7 18 . 20 . 22 .....
D =  1 3 1 3 5 7 5 7  5 7  5 7  9 11 9 11
S becomes:
S = 2 1 4 3 6 1 8 3 10 5 12 7 14 5 16 7 18 5 20 7 22 5 24 7 26 9 28 11 30 9 32 11 34 13 36 15 38 13 40....

We can start S with '1' and affirm now that "the absolute first differences of S are the odd terms of S":
S = 1 2 1 4 3 6 1 8 3 10 5 12 7 14 5 16 7 18 5 20 7 22 5 24 7 26 9 28 11 30 9 32 11 34 13 36 15 38 13 40....

More terms? Pattern?
Best,
É.

```