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A001868 Number of n-bead necklaces with 4 colors.
(Formerly M3390 N1370)
19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
From Richard L. Ollerton, May 07 2021: (Start)
Here, as in A000031, turning over is not allowed.
(1/n) * Dirichlet convolution of phi(n) and 4^n, n>0. (End)
REFERENCES
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).
LINKS
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
FORMULA
a(n) = (1/n)*Sum_{d|n} phi(d)*4^(n/d) = A054611(n)/n, n>0.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 4*x^n)/n. - Herbert Kociemba, Nov 01 2016
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
MAPLE
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;
MATHEMATICA
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 *)
PROG
(PARI) a(n) = if (n, sumdiv(n, d, eulerphi(d)*4^(n/d))/n, 1); \\ Michel Marcus, Nov 01 2016
CROSSREFS
Column 4 of A075195.
Cf. A054611.
Sequence in context: A212330 A291412 A366645 * A217696 A223014 A038783
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified February 26 11:46 EST 2024. Contains 370352 sequences. (Running on oeis4.)