A045610 Number of different energy states of n positive and n negative charges on a string.

1, 1, 3, 7, 22, 70, 249, 880, 3238, 12180, 46247, 174458, 672920, 2585414, 10015955

Offset

0,3

Links

Table of n, a(n) for n=0..14.

Sean A. Irvine, Java program, GitHub.

Example

For n=3 the 20 different arrangements of -1,-1,-1,1,1,1 result in 7 energy levels (sum of signed inverse distances):
{0,0,0,1,1,1},{1,1,1,0,0,0}: 13/10
{0,0,1,0,1,1},{1,1,0,1,0,0}: -41/30
{0,0,1,1,0,1},{0,1,0,0,1,1},{1,0,1,1,0,0},{1,1,0,0,1,0}: -56/30
{0,0,1,1,1,0},{0,1,1,1,0,0},{1,0,0,0,1,1},{1,1,0,0,0,1}: -8/10
{0,1,0,1,0,1},{1,0,1,0,1,0}: -37/10
{0,1,0,1,1,0},{0,1,1,0,1,0},{1,0,0,1,0,1},{1,0,1,0,0,1}: -89/30
{0,1,1,0,0,1},{1,0,0,1,1,0}: -71/30
so the multiplicities are 4*2 + 3*4 = 20 = binomial(6,3).

Mathematica

f[li_: {(0 | 1) ..}] := Outer[Times, 2 li - 1, 2 li - 1];
Table[Length @ Tally[Total[1/DeleteCases[f[#] DistanceMatrix[Range[2 n]], 0, 2], 2] & /@ Permutations[Join[Table[0, n], Table[1, n]]]], {n, 10}] (* Wouter Meeussen, Mar 15 2021 *)

Crossrefs

Cf. A045723.

Sequence in context: A319312 A325213 A148688 * A259809 A340022 A181769

Adjacent sequences: A045607 A045608 A045609 * A045611 A045612 A045613

Keyword

nonn,more

Author

Wouter Meeussen

Extensions

Corrected and extended by Wouter Meeussen, Mar 15 2021

a(12)-a(15) from Sean A. Irvine, Mar 15 2021

Status

approved