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.