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

Author

N. J. A. Sloane, Apr 16 2000

Keywords

• nonn

Examples

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

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

Crossrefs

Column 8 of A075195.

Column k=1 of A184294.

Cf. A054615.

Programs

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 *)

PARI

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

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

Extensions

Edited by Christian G. Bower, Sep 07 2002

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