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A001868
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Number of n-bead necklaces with 4 colors.
(Formerly M3390 N1370)
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19
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1, 4, 10, 24, 70, 208, 700, 2344, 8230, 29144, 104968, 381304, 1398500, 5162224, 19175140, 71582944, 268439590, 1010580544, 3817763740, 14467258264, 54975633976, 209430787824, 799645010860, 3059510616424, 11728124734500, 45035996273872, 173215372864600, 667199944815064
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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 4^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)*4^(n/d) = A054611(n)/n, n>0.
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 4^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 4^(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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A001868 := proc(n) local d, s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*4^(n/d); od; RETURN(s/n); fi; end;
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MATHEMATICA
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a[n_] := (1/n)*Total[ EulerPhi[#]*4^(n/#) & /@ Divisors[n]]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Oct 21 2011 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-4*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 01 2016 *)
k=4; 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, sumdiv(n, d, eulerphi(d)*4^(n/d))/n, 1); \\ Michel Marcus, Nov 01 2016
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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