[seqfan] Pairs and first differences

[Eric Angelini]

Hello SeqFans,
See S as a succession of pairs of integers (a,b),(c,d),(e,f),... 
Compute the absolute differences in each pair. 
The succession of such differences is S itself.

S=1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,...

S was build starting with a(1)=1 
and extended with the smallest 
integer not yet present in S and
not leading to a contradiction.
S is thus a permutation of N 
(integers > 0).

If we want S to be strictly increasing we get T:

T=1,2,3,5,6,9,10,15,16,22,23,32,33,43,44,59,60,76,77,99,...

U is nice (?); forgetting the pairs
and considering the absolute differences
only, we get U again, but with every
term repeated:

U=1,2,3,5,7,10,13,18,22,29,36,46,56,...

The abs diff are:
1,1,2,2,3,3,5,5,7,7,10,10,...

Now the challenge:
U was meant strictly increasing. But if we drop this condition, is it
possible to build V which would be
a permutation of N?

V=1,2,3,5,7,10,13,18,23,16,9,19,29,...

The abs diff are again:
1,1,2,2,3,3,5,5,7,7,10,10,...

Best,
É.