A386388 a(n) is the number of complement pairs of 2n-bead balanced bicolor necklaces.

0, 0, 0, 1, 3, 11, 36, 118, 395, 1337, 4598, 15986, 56270, 199854, 716132, 2584754, 9391051, 34315811, 126040590, 465062362, 1723070794, 6407806952, 23910175804, 89493721076, 335912391966, 1264105728842, 4768446886764, 18027215662284, 68291878325138

Offset: 0

Author

Tilman Piesk, Jul 20 2025

Keywords

• nonn

Comments

A003239(n) is the number of 2n-bead balanced bicolor necklaces, and A000013(n) is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.

Doubled entries: 0, 0, 0, 2, 6, 22, 72, 236, 790, 2674, 9196, 31972, 112540, 399708, 1432264, ...

Examples

			  n  | A003239(n) A000013(n) | 2*a(n)      a(n)
  0  |         1          1  |     0         0
  1  |         1          1  |     0         0
  2  |         2          2  |     0         0
  3  |         4          2  |     2         1
  4  |        10          4  |     6         3
  5  |        26          4  |    22        11
  6  |        80          8  |    72        36
  7  |       246         10  |   236       118
  8  |       810         20  |   790       395
  9  |      2704         30  |  2674      1337
 10  |      9252         56  |  9196      4598
Examples for n=4 with necklaces of length 8:
A000013(4) = 4 necklaces are self-complementary:
 00001111, 00110011, 01010101, 00101101 (compare A385665)
There are a(n) = 3 pairs of complementary necklaces:
 (00110101, 00101011), (00100111, 00011011), (00010111, 00011101)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1670

Crossrefs

Cf. A000013, A003239, A385665.

Programs

Mathematica

a[0]=0;a[n_]:=( Sum[ EulerPhi[n/k]*Binomial[2k, k]/(2n), {k, Divisors[n]}]- Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 0, Divisors[n]])/2;Array[a,29,0] (* James C. McMahon, Jul 30 2025 *)

Formula

a(n) = (A003239(n) - A000013(n)) / 2.