[seqfan] Successive weighing scales
Eric Angelini
Eric.Angelini at kntv.be
Mon Jun 14 16:27:52 CEST 2010
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.
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