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 A054627 Number of n-bead necklaces with 8 colors. 5
 1, 8, 36, 176, 1044, 6560, 43800, 299600, 2097684, 14913200, 107377488, 780903152, 5726645688, 42288908768, 314146329192, 2345624810432, 17592187093524, 132458812569728, 1000799924679192, 7585009898729264, 57646075284033552, 439208192231379680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Necklace. 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

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Last modified September 20 17:41 EDT 2021. Contains 347587 sequences. (Running on oeis4.)