A054627 Number of n-bead necklaces with 8 colors.

1, 8, 36, 176, 1044, 6560, 43800, 299600, 2097684, 14913200, 107377488, 780903152, 5726645688, 42288908768, 314146329192, 2345624810432, 17592187093524, 132458812569728, 1000799924679192, 7585009898729264, 57646075284033552, 439208192231379680

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Necklace.

Index entries for sequences related to necklaces

Formula

a(n) = (1/n)*Sum_{d|n} phi(d)*8^(n/d), n > 0.

G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 8*x^n)/n. - Herbert Kociemba, Nov 02 2016

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 8^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021

Example

G.f. = 1 + 8*x + 36*x^2 + 176*x^3 + 1044*x^4 + 6560*x^5 + 43800*x^6 + ...

Maple

with(combstruct):A:=[N, {N=Cycle(Union(Z$8))}, unlabeled]: seq(count(A, size=n), n=0..20); # Zerinvary Lajos, Dec 05 2007

Mathematica

mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-8*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=8; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *)

Prog

(PARI) a(n)=if(n==0, 1, 1/n*sumdiv(n, d, eulerphi(d)*8^(n/d))); \\ Altug Alkan, Sep 21 2018

Crossrefs

Column 8 of A075195.

Column k=1 of A184294.

Cf. A054615.

Sequence in context: A164545 A199310 A316107 * A019022 A079819 A238815

Adjacent sequences: A054624 A054625 A054626 * A054628 A054629 A054630

Keyword

nonn

Author

N. J. A. Sloane, Apr 16 2000

Extensions

Edited by Christian G. Bower, Sep 07 2002

a(0) corrected by Herbert Kociemba, Nov 02 2016

Status

approved