A045611 - OEIS

A045611

Number of different energy states of n positive and n negative charges on a necklace. Different sets of distances between n points chosen from 2n equally spaced points on a circle.

%I #15 Mar 17 2021 18:55:01

%S 1,1,2,3,7,13,35,85,254,701,2377,7944,25220,95910,332300,1164825,

%T 4379920,16649851,58473414

%N Number of different energy states of n positive and n negative charges on a necklace. Different sets of distances between n points chosen from 2n equally spaced points on a circle.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a045/A045611.java">Java program</a> (github)

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

%e a(10) = 2377 because the multiplicities {1; 2; 5; 10; 20; 40; 60; 80; 120} occur {1; 1; 3; 22; 362; 1855; 1; 130; 2} times and 2377 = 1 + 1 + 3 + 22 + 362 + 1855 + 1 + 130 + 2. Total number of configurations = 1*1 + 1*2 + 3*5 + 22*10 + .. = 92378 (see C(2n + 1, n + 1)=A001700).

%t n=4; perm=Permutations[ Table[ -1, {n-1} ]~Join~Table[ 1, {n} ] ]; res=Table[ (Prepend[ #, -1 ].RotateRight[ Prepend[ #, -1 ], k ])/2, {k, 1, n} ]&/@perm; kort=Length[ Length/@Split[ Sort[ res ] ] ]

%Y For n=0..7 this is equal to A006840. Cf. A045610, A045611, A045612, A006840, A077078, A077079.

%K nonn,more

%O 0,3

%A _Wouter Meeussen_

%E a(15) corrected and a(16)-a(18) from _Sean A. Irvine_, Mar 17 2021