# [seqfan] Pairs and differences reproduce S

Eric Angelini Eric.Angelini at kntv.be
Sat Feb 1 23:36:42 CET 2014

```Hello SaqFans,
here is a permutation of the Naturals not yet in the OEIS, I think:

S=1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,15,27,17,26,18,34,20,33,23,42,25,35,24,45,...

Put parentheses around each pair of integers, like this:

S=(1,2),(3,5),(4,7),(6,11),(8,12),(9,16),(13,19),(10,21),(14,22),(15,27),(17,26),(18,34),(20,33),(23,42),(25,35),(24,45),...

If you replace each pair of integers by their difference, you'll get S again:

S=1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,15,27,17,26,18,34,20,33,23,42,25,35,24,45,...

S was always extended with the smallest unused term not leading to a contradiction.

For example, after:

S=(1,2),(3,5),(4,7),(6,11),(8,12),(9,16),...

... one cannot extend S with 10 as 10 would produce the pair (10,16) with a "16" being already used.

... similarily, after:

S=(1,2),(3,5),(4,7),(6,11),(8,12),(9,16),(13,19),(10,21),(14,22),(15,27),(17,26),(18,34),(20,33),(23,42),...

... one cannot extend S with 24 as 24 would produce the pair (24,34) with a "34" being already used.

Best,
É.

Catapulté de mon aPhone
```