[seqfan] A fun seq for Zakir (first diff of odd rank)

[Eric Angelini]

Hello SeqFans,

I've had the idea of a self-descriptive seq:

«The first differences of odd rank are the sequence S itself»

If S must be monotonically increasing we get:

     S = 1 2 3 5 6 9 10 15 16 22 23 32 33  43 44  59 60  76 77  99 100  123 ... 
1st diff: 1 1 2 1 3 1  5  1  6  1  9  1  10  1  15  1  16  1  22  1   23
odd diff: 1 . 2 . 3 .  5  .  6  .  9  .  10  .  15  .  16  .  22  .   23  .

... as one can see, odd rank diff are S itself.

Now the difficult part:

After seing S one could wonder if, dropping the increasing 
constraint, it would be possible for T to be a permutation
of the Naturals (1,2,3,4,5,6,7,8,...)

I think it is possible, yes -- but T is a nightmare to construct:

     T = 1 2 3 5 4 7 6 11 8 12 9 16 13 19 10  21 14 22 15  27  17 26 18  34 ...
1st diff: 1 1 2 1 3 1 5  3 4  3 7  3  6  9  11  7  8  7  12  10  9  8  16
odd diff: 1 . 2 . 3 . 5  . 4  . 7  .  6  .  11  .  8  .  12   .  9  .  16

... as one can see again, odd rank diff are T itself -- and no term
is a copy of a previous one (first missing terms are 20,23,24,25,28,...)

But how was constructed T (I hope I made no mistake)?

I wanted always the new (free) term of T to be _the smallest term
not yet present in T_ (as always); but one has to be carefull:

     T = 1 2 3 5 4 7 6 11 8 12 9 16 
1st diff: 1 1 2 1 3 1 5  3 4  3 7   
odd diff: 1 . 2 . 3 . 5  . 4  . 7  

... now we see that the _smallest term not yet present in T_ could
be 10:

     T = 1 2 3 5 4 7 6 11 8 12 9 16 10
1st diff: 1 1 2 1 3 1 5  3 4  3 7  6  
odd diff: 1 . 2 . 3 . 5  . 4  . 7  .

But 10 leads to an impossibility because of the "odd diff" line
which imposes "6" as the next "first diff":

     T = 1 2 3 5 4 7 6 11 8 12 9 16 10
1st diff: 1 1 2 1 3 1 5  3 4  3 7  6  6
odd diff: 1 . 2 . 3 . 5  . 4  . 7  .  6

... and we have no available integer to expand T: 10+6=16 already taken
                                               or 10-6= 4 already taken

So we must discard 10 and try the next _smallest term not yet present
in T_ which is "13":

     T = 1 2 3 5 4 7 6 11 8 12 9 16 13 19  a   b  c  d  e   f   g  h  i   j ... 
1st diff: 1 1 2 1 3 1 5  3 4  3 7  3  6  ?  11  ?  8  ?  12   ?  9  ?  16
odd diff: 1 . 2 . 3 . 5  . 4  . 7  .  6  .  11  .  8  .  12   .  9  .  16 

If this is of interest, could someone have a try on S and T?
Best,
É.