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A054629 Number of n-bead necklaces with 10 colors. 3
1, 10, 55, 340, 2530, 20008, 166870, 1428580, 12501280, 111111340, 1000010044, 9090909100, 83333418520, 769230769240, 7142857857190, 66666666680272, 625000006251280, 5882352941176480, 55555555611222370, 526315789473684220 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Eric Weisstein's World of Mathematics, Necklace.
FORMULA
a(n) = (1/n)*Sum_{d|n} phi(d)*10^(n/d) = A054617(n)/n, n > 0.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 10*x^n)/n. - Herbert Kociemba, Nov 02 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 10^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021
EXAMPLE
G.f. = 1 + 10*x + 55*x^2 + 340*x^3 + 2530*x^4 + 20008*x^5 + 166870*x^6 + ...
MAPLE
with(combstruct):A:=[N, {N=Cycle(Union(Z$10))}, unlabeled]: seq(count(A, size=n), n=0..19); # Zerinvary Lajos, Dec 05 2007
MATHEMATICA
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-10*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=10; 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)*10^(n/d))); \\ Altug Alkan, Sep 21 2018
CROSSREFS
Column 10 of A075195.
Cf. A054617.
Sequence in context: A316109 A188168 A199413 * A290359 A030114 A001557
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 April 14 07:36 EDT 2024. Contains 371655 sequences. (Running on oeis4.)