**The first differences of S**

are the terms in even
position in S

S = 2,1,3,2,5,1,6,3,9,4,11,5,13,3,12,6,17,5,18,7,21,6,22,8,25,10,...

Even pos. E = . 1 . 2 . 1 .
3 . 4 . 5 . 3 . 6 . 5 . 7 . 6 . 8 . 10

First dif **D**
= 1 2 1 3 4 5 3
6 5 7 6 8 10 ... --> this forms E
again

So, generally speaking, we have:

S = a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z, ...

E = b
d f h
j l n
p r t
v x z ...

**D**
= b d f h j l n p r t v x z ...

To start S we have to be sure that a=b+b; this happens with the “smallest available difference
b” which is b=1. This forces a=2 and we have the first
two terms of S:

S =
2,1,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z, ...

**D**
= 1 d f h j l n p r t v x z ...

The next step is to compute d (on the **D** line). Should we decide to always
take “the smallest possible difference” would give d=1 thus c=2, giving us the
third and fourth terms of S:

S =
2,1,2,1,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z, ...

**D**
= 1 1 f h j l
n p r t v x z ...

It follows that f=1:

S = 2,1,2,1,e,1,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,
...

**D**
= 1 1 1 h j l n p r t v x z ...

Should we decide again to take “the
smallest possible difference” would give h=1 and e=2:

S =
2,1,2,1,2,1,g,1,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z, ...

**D**
= 1 1 1 1 j l n p r t v x z ...

... etc. This
way of building S would produce the rather dull (but correct) sequence:

S =
2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1, ...

**D** = 1 1 1 1 1 1 1 1 1 1 1 1 1 ...

... where,
indeed, “the first differences of S are the terms in even position of S
itself”.

So, instead of choosing systematically 1
for the next difference on line D, we have decided here, __when we had the
choice__, to take always “the smallest possible difference __not yet in D
and not leading to a contradiction__”:

S = 2,1,3,2,5,1,6,3,9,4,11,5,13,3,12,6,17,5,18,7,21,6,22,8,25,10,...

First dif **D**
= 1 2 1 3 4 5 3
6 5 7 6 8 10 ... (those are the terms in
even position in S)

The terms where we had the
choice are in cyan color hereunder (note that when we had the choice we
never did repeat an existing term):

S = 2,1,3,2,5,1,6,3,9,4,11,5,13,3,12,6,17,5,18,7,21,6,22,8,25,10,...

First dif **D**
= 1 2 1 3 4 5 3 6 5 7 6 8 10 9 6 11 ... (those are the terms in even position in S)

So, this is the
“lexicographically first interesting sequence S, where the absolute first
differences of S are the terms in even position in S itself”.

If this is of interest, could someone
please compute a 100 terms or so – and submit S to the OEIS?

Best,

É.

__________

P.-S.

The “lexicographically first interesting
sequence T, where the absolute first differences of T are the terms in odd
position in T itself” is there.