OFFSET
1,1
LINKS
C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
M. Taniguchi, H. Du, and J. S. Lindsey, Enumeration of virtual libraries of combinatorial modular macrocyclic (bracelet, necklace) architectures and their linear counterparts, Journal of Chemical Information and Modeling, 53 (2013), 2203-2216.
FORMULA
"DIK" (bracelet, indistinct, unlabeled) transform of 4, 0, 0, 0, ...
a(n) = A081720(n,4), n >= 4. - Wolfdieter Lang, Jun 03 2012
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 4*x^n)/n + (1+4*x+6*x^2)/(1-4*x^2))/2. - Herbert Kociemba, Nov 02 2016
a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))* Sum_{d|n} phi(d)*k^(n/d), where k=4 is the maximum number of colors. - Robert A. Russell, Sep 24 2018
a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i), where k=4 is the maximum number of colors. (See A075195 formulas.) - Richard L. Ollerton, May 04 2021
EXAMPLE
For n=2, the ten bracelets are AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD. - Robert A. Russell, Sep 24 2018
MATHEMATICA
mx=40; CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-4*x^n]/n, {n, mx}]+(1+4 x+6 x^2)/(1-4 x^2))/2, {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
k=4; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved