

A032169


Number of aperiodic necklaces of n beads of 2 colors, 11 of them black.


2



1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32065, 58786, 104006, 178296, 297160, 482885, 766935, 1193010, 1820910, 2731365, 4032015, 5864749, 8414640, 11920740, 16689036, 23107896, 31666376, 42975796
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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 nk of them are white, then a_k(n) = (1/k)*Sum_{dgcd(n,k)} mu(d)*binomial(n/d  1, k/d  1) = (1/n)*Sum_{dgcd(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(n1, k1)/k). In such a case, the sequence becomes a column for triangle A011847. (This is not true when k is composite >= 4.)
(End)


LINKS

Table of n, a(n) for n=12..39.
C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to Lyndon words


FORMULA

"CHK[ 11 ]" (necklace, identity, unlabeled, 11 parts) transform of 1, 1, 1, 1, ...
G.f.: (x^11/11)*(1/(1x)^111/(1x^11)).  Herbert Kociemba, Oct 16 2016
a(n) = (1/11)*(binomial(n1, 10)  I(11n)) = floor(binomial(n1, 10)/11) for n >= 12, where I(ab) = 1 if integer a divides integer b, and 0 otherwise.  Petros Hadjicostas, Aug 26 2018


MATHEMATICA

CoefficientList[Series[x^11/11 (1/(1x)^111/(1 x^11)), {x, 0, 50}], x] (* Herbert Kociemba, Oct 16 2016 *)


CROSSREFS

A column of triangle A011847.
Sequence in context: A060101 A036422 A166214 * A032196 A011780 A036631
Adjacent sequences: A032166 A032167 A032168 * A032170 A032171 A032172


KEYWORD

nonn


AUTHOR

Christian G. Bower


STATUS

approved



