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A001867
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Number of n-bead necklaces with 3 colors.
(Formerly M2548 N1008)
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24
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1, 3, 6, 11, 24, 51, 130, 315, 834, 2195, 5934, 16107, 44368, 122643, 341802, 956635, 2690844, 7596483, 21524542, 61171659, 174342216, 498112275, 1426419858, 4093181691, 11767920118, 33891544419, 97764131646, 282429537947, 817028472960, 2366564736723
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OFFSET
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0,2
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COMMENTS
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Here, as in A000031, turning over is not allowed.
(1/n) * Dirichlet convolution of phi(n) and 3^n, n>0. (End)
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 162.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).
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LINKS
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FORMULA
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a(n) = (1/n)*Sum_{d|n} phi(d)*3^(n/d), n>0.
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 3^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 3^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
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MAPLE
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with(numtheory): A001867:= n-> `if` (n=0, 1, add (phi(d)* 3^(n/d), d=divisors(n))/n): seq (A001867(n), n=0..40);
spec := [N, {N=Cycle(bead), bead=Union(R, G, B), R=Atom, B=Atom, G=Atom}]; [seq(combstruct[count](spec, size=n), n=1..40)];
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MATHEMATICA
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Prepend[Table[CyclicGroupIndex[n, t]/.Table[t[i]->3, {i, 1, n}], {n, 1, 28}], 1] (* Geoffrey Critzer, Sep 16 2011 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-3*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 01 2016 *)
k=3; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *)
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PROG
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(PARI) a(n)=if (n==0, 1, 1/n * sumdiv(n, d, eulerphi(d)*3^(n/d) )); /* Joerg Arndt, Jul 04 2011 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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