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 3^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
T. D. Noe, Table of n, a(n) for n = 0..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
V. E. Hoggatt, The Fifth Oldie from the Vault. Problem B-415, Elementary Problems and Solutions, Fibonacci Quart. 59 (2021), no. 3, pp. 274-275.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 3
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.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
FORMULA
a(n) = (1/n)*Sum_{d|n} phi(d)*3^(n/d), n>0.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 3*x^n)/n). - Herbert Kociemba, Nov 01 2016
a(n) ~ 3^n/n. - Vaclav Kotesovec, Nov 01 2016
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
MAPLE
MATHEMATICA
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 *)
PROG
(PARI) a(n)=if (n==0, 1, 1/n * sumdiv(n, d, eulerphi(d)*3^(n/d) )); /* Joerg Arndt, Jul 04 2011 */
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved