[SeqFan] close to A006840
Wouter Meeussen
eu000949 at pophost.eunet.be
Thu Aug 6 20:43:49 CEST 1998
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
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