OFFSET
12,2
COMMENTS
From Petros Hadjicostas, Aug 26 2018: (Start)
Assume n >= k >= 2. If a_k(n) is the number of aperiodic necklaces of n beads of 2 colors such that k of them are black and n-k of them are white, then a_k(n) = (1/k)*Sum_{d|gcd(n,k)} mu(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} mu(d)*binomial(n/d, k/d). This follows from Herbert Kociemba's general formula for the g.f. of (a_k(n): n>=1) that can be found in the comments for sequence A032168.
For k prime, we get a_k(n) = floor(binomial(n-1, k-1)/k). In such a case, the sequence becomes a column for triangle A011847. (This is not true when k is composite >= 4.)
(End)
LINKS
Ray Chandler, Table of n, a(n) for n = 12..1012
C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1,1,-10,45,-120,210,-252,210,-120,45,-10,1).
FORMULA
"CHK[ 11 ]" (necklace, identity, unlabeled, 11 parts) transform of 1, 1, 1, 1, ...
G.f.: (x^11/11)*(1/(1-x)^11-1/(1-x^11)). - Herbert Kociemba, Oct 16 2016
a(n) = (1/11)*(binomial(n-1, 10) - I(11|n)) = floor(binomial(n-1, 10)/11) for n >= 12, where I(a|b) = 1 if integer a divides integer b, and 0 otherwise. - Petros Hadjicostas, Aug 26 2018
MATHEMATICA
CoefficientList[Series[x^11/11 (1/(1-x)^11-1/(1- x^11)), {x, 0, 50}], x] (* Herbert Kociemba, Oct 16 2016 *)
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
STATUS
approved