A081721 Number of bracelets of n beads in up to n colors.

1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356, 24330595997127372497, 1092955780817066765469, 52063675152021153895330, 2621440000054016000176044

Offset: 1

Author

N. J. A. Sloane, based on information supplied by Gary W. Adamson, Apr 05 2003

Keywords

• nonn

Comments

T(n,n), T given in A081720.

From Olivier Gérard, Aug 01 2016: (Start)

Number of classes of functions of [n] to [n] under rotation and reversal.

.

Classes can be of size between 1 and 2n

depending on divisibility properties of n.

.

n 1 2 3 4 5 n 2n

----------------------------------------

1 1

2 2 1

3 3 0 6 1

4 4 6 0 30 15

5 5 0 0 120 252

6 6 15 30 725 3515

7 7 0 0 2394 57627

.

(End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000312 All endofunctions

Cf. A000169 Classes under translation mod n

Cf. A001700 Classes under sort

Cf. A056665 Classes under rotation

Cf. A168658 Classes under complement to n+1

Cf. A130293 Classes under translation and rotation

Cf. A275549 Classes under reversal

Cf. A275550 Classes under reversal and complement

Cf. A275551 Classes under translation and reversal

Cf. A275552 Classes under translation and complement

Cf. A275553 Classes under translation, complement and reversal

Cf. A275554 Classes under translation, rotation and complement

Cf. A275555 Classes under translation, rotation and reversal

Cf. A275556 Classes under translation, rotation, complement and reversal

Cf. A275557 Classes under rotation and complement

Cf. A275558 Classes under rotation, complement and reversal

Row sums of partition array A213941.

Main diagonal of A321791.

Programs

Mathematica

Table[CycleIndex[DihedralGroup[n],s]/.Table[s[i]->n,{i,1,n}],{n,1,20}] (* Geoffrey Critzer, Jun 18 2013 *)
t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); a[n_] := t[n, n]; Array[a, 20] (* Jean-François Alcover, Nov 02 2017, after Maple code for A081720 *)

Formula

a(n) ~ n^(n-1) / 2. - Vaclav Kotesovec, Mar 18 2017

Extensions

Name changed by Olivier Gérard, Aug 05 2016

Name revised by Álvar Ibeas, Apr 20 2018