The first differences of S

are the odd terms of S

We start alternating even numbers (in yellow) 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....

Best,

É.