A054625 Number of n-bead necklaces with 6 colors.
1, 6, 21, 76, 336, 1560, 7826, 39996, 210126, 1119796, 6047412, 32981556, 181402676, 1004668776, 5597460306, 31345666736, 176319474366, 995685849696, 5642220380006, 32071565263716, 182807925027504, 1044616697187576, 5982804736593846
Offset: 0
Keywords
Examples
G.f. = 1 + 6*x + 21*x^2 + 76*x^3 + 336*x^4 + 1650*x^5 + 7826*x^6 + 39996*x^7 + ...
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..1289
- Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 22.
- Eric Weisstein's World of Mathematics, Necklace.
- Index entries for sequences related to necklaces
Programs
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Maple
with(combstruct):A:=[N,{N=Cycle(Union(Z$6))},unlabeled]: seq(count(A,size=n),n=0..22); # Zerinvary Lajos, Dec 05 2007 -
Mathematica
f[n_] := Block[{d = Divisors@ n}, Total[EulerPhi[d]*6^(n/d)]/n]; f[0] = 1; Array[f, 23, 0] (* Robert G. Wilson v, Jan 01 2013 *) mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-6*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
Formula
a(n) = (1/n)*Sum_{d|n} phi(d)*6^(n/d), n > 0.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 6*x^n)/n. - Herbert Kociemba, Nov 02 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 6^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021
Extensions
Edited by Christian G. Bower, Sep 07 2002
a(0) corrected by Herbert Kociemba, Nov 02 2016