[SeqFan] close to A006840

[Wouter Meeussen]

2n charges in a ring configuration in the plane,
n positive (+1) and n negative (-1) ones, equally spaced.
count the number of potential energy levels.

pro memori:
A006840 1,2,3,7,13,35,85,257,765,2518
configurations :new =

        1,2,3,7,13,35,85,254,701,2377

Now I'm confident, having the following consistency check:

2n    Energy  # of      (nr k of E-levels with degeneracy d)
      levels states     (k1*d1+k2*d2+...)
2       1       1       1*1
4       2       3       1*1+1*2
6       3       10      1*1+1*3+1*6
8       7       35      1*1+1*2+2*4+3*8
10      13      126     1*1+3*5+7*10+2*20
12      35      462     1*1+1*2+1*3+4*6+20*12+8*24
14      85      1716    1*1+7*7+35*14+42*28
16      254     6435    1*1+1*2+2*4+11*8+88*16+148*32+3*64
18      701     24310   1*1+1*3+1*6+14*9+136*18+488*36+9*54+51*72
20      2337    92378   1*1+1*2+3*5+22*10+362*20+1855*40+1*60+130*80+2*120

it is aparent that the degeneracy is built from divisors or multiples of (2n).
The # of states is of course (n choose (n/2))/2 , half the central binomial
coefficients.


%I A000000 
%S A000000 1,2,3,7,13,35,85,254,701,2377
%N A000000 number of configurations, excluding rotation, reflection and black-
white interchange, of n black and n white beads on a necklace
%R A000000 
%A A000000 w.meeussen.vdmcc at vandemoortele.be
%O A000000 0,2
%K A000000 nonn

The number of states would double if we dropped the black-white interchange
(or the
symmetry between negative and positive charges).
the next entry (22 beads) would run for approx 4 days on a 90 MHz machine.

Some combinatorial intelligence should take over here. That's not me.


wouter.
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be