Suggestion for a sequence: weights on a circle

[Hugo Pfoertner]

Hugo Pfoertner schrieb:
> 
> Now knowing that perfectly balanced arrangements of n weights will not
> be possible if n is a prime power, one could ask the more general
> question: How many solutions are there for a given n that minimize the
> remaining imbalance?
> 
> With a slightly modified program I calculated:
> (the floating point number gives the distance of the center of gravity
> from (0,0)
> 
>  Number of weights: 3
>  0.577350    1    1  2  3
>  0.577350    2    1  3  2
> 
>  Number of weights: 4
>  0.353553    1    1  3  2  4
>  0.353553    2    1  4  2  3
> 
>  Number of weights: 5
>  0.089806    1    1  4  3  2  5
>  0.089806    2    1  5  2  3  4
> 
>  Number of weights: 7
>  0.010927    1    1  4  7  2  3  5  6
>  0.010927    2    1  6  5  3  2  7  4
> 
>  Number of weights: 8
>  0.016415    1    1  4  7  3  6  2  5  8
>  0.016415    2    1  7  4  3  6  5  2  8
>  0.016415    3    1  8  2  5  6  3  4  7
>  0.016415    4    1  8  5  2  6  3  7  4
> 
>  Number of weights: 9
>  0.003184    1    1  5  9  2  7  3  4  8  6
>  0.003184    2    1  5  9  4  2  6  7  3  8
>  0.003184    3    1  6  5  4  9  2  3  7  8
>  0.003184    4    1  6  8  4  3  7  2  9  5
>  0.003184    5    1  8  3  7  6  2  4  9  5
>  0.003184    6    1  8  7  3  2  9  4  5  6
> 
>  Number of weights: 11
>  0.000019    1    1  8  9  5  2  6 10  7  3  4 11
>  0.000019    2    1 11  4  3  7 10  6  2  5  9  8
> 
>  Number of weights: 13
>  0.000028    1    1  2  7 12 13  4  5  3  8  6 11  9 10
>  0.000028    2    1  4 11  6  5 12  7  2  9  8  3 10 13
   ^^^^^^^^ should be 0.000039652 accurate distance
>  0.000028    3    1  5  3  8 12 10  7  4  2  6 11  9 13
   ^^^^^^^^ should be 0.000062547
>  0.000028    4    1 10  9 11  6  8  3  5  4 13 12  7  2 
>  0.000028    5    1 13  9 11  6  2  4  7 10 12  8  3  5
   ^^^^^^^^ should be 0.000062547
>  0.000028    6    1 13 10  3  8  9  2  7 12  5  6 11  4
   ^^^^^^^^ should be 0.000039652

Unfortunately the criterion I had chosen to accept two non-perfect
solutions as being equivalent was not sharp enough for n>12. With a
reduced acceptance limit I get only two solutions for n=13:

 0.000028458    1    1  2  7 12 13  4  5  3  8  6 11  9 10
 0.000028458    2    1 10  9 11  6  8  3  5  4 13 12  7  2

For n=16 my program finds 288 equivalent best possible solutions:

http://www.randomwalk.de/sequences/balcir16.txt

The lexicographically first config is:

 0.000009114   [1  3  5 13 16  7 10  2 14  4  6  9 12  8 11 15]

I hope to get results for n=17 next week. The required accumulated CPU
time will be ~= 36 days.

> 
> Merging this with the numbers of perfectly balanced solutions we can
> make the following sequence, starting at n=3:
>

Corrected:

 2 2 2 4 2 4 6 48 2 1464 2 1440 96 288
 
Hugo Pfoertner