[seqfan] Successive weighing scales

[Eric Angelini]

Hello SeqFans,

Here is a first succession of empty weighing scales:

|     |  |     |  |     |  |     |  |     |  |     |  |     |  |     |  |     | 
+--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+ 

We write under each scale its unbalance:

|     |  |     |  |     |  |     |  |     |  |     |  |     |  |     |  |     | 
+--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+ 
   0        0        0        0        0        0        0        0        0   

We fill each scale with two integer weights:

|1   2|  |3   5|  |6   9|  |10 15|  |16 22|  |23 32|  |33 43|  |44 59|  |60 76| 
+--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+  +--.--+ 
   1        2        3        5        6        9        10       15       16   

Ok, you get it, the unbalances's seq is the seq formed by the successive weights:

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

S is monotocally increasing and not in the OEIS.

If we drop the "monotocally increasing" constraint and want the sequence to
be a permutation of the Natural numbers (1,2,3,4,5,6,7,...n) we have:

T = 1,2,3,5,4,7,6,11,8,12,9,16,13,19,10,21,14,22,15,27,17,26,18,34,20,33,23,42,25,35,24,45,29,43,28,50,...

The algorithm used here was, as usual, "use the smallest available integer 
not yet present in T and not leading to a contradiction". T is not in the
OEIS either.

Building T is smooth -- except for some weights which have to be delayed:

T = 1,2,3,5,4,7,6,11,8,12,9,16,13,19, ... is ok
     1   2   3   5    4    7     6

T = 1,2,3,5,4,7,6,11,8,12,9,16,10,--, ... is not (10-6 and 10+6 are already in T)
     1   2   3   5    4    7     6

Now, what happens _between_ the weighing scales?

For S, the scales are always "separated" by weights of 1 unit:

S = |1   2|   |3   5|   |6   9|   |10 15|   |16 22|   |23 32|   |33 43|   |44 59|   |60 76| 
    +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 1 +--.--+ 
       1    ^    2    ^    3    ^    5    ^    6    ^    9    ^    10   ^    15   ^    16   

Could the succession of the separations be the sequence itself? Let's try:

S' = |1   2|   |3   5|   |7  10|   |13 18|   |23 30|   |37 47|    |57 70|    |83  101|   
     +--.--+ 1 +--.--+ 2 +--.--+ 3 +--.--+ 5 +--.--+ 7 +--.--+ 10 +--.--+ 13 +---.---+  ...
        1    ^    2    ^    3    ^    5    ^    7    ^    10   ^^    13   ^^     18         

It works... but S' is already in the OEIS:
http://www.research.att.com/~njas/sequences/A033485
"a(n) = a(n-1) + a([n/2]), a(1) = 1"

Now the difficult part: could we build a sequence similar to S, but dropping the
"monotonically increasing" constraint?

We are thus looking for a sequence T' where:
- a(n) is not always > a(n-1)
- a(n) doesn't show twice
- the succession of the separations (between successive scales) form T' itself

I think T' is not impossible to construct and might start like this:

T' = |1   2|   |3   5|   |7  10|   |13 18|   |23 16|   |9  19|    |29 42|    |55 37|   
     +--.--+ 1 +--.--+ 2 +--.--+ 3 +--.--+ 5 +--.--+ 7 +--.--+ 10 +--.--+ 13 +--.--+  ...
        1    ^    2    ^    3    ^    5    ^    7    ^    10   ^^    13   ^^    18         

Could T' be a permutation of the Naturals? Mmmmmh...

Best,
É.