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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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=12..39.

C. G. Bower, Transforms (2)

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

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/(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

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

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Last modified October 19 09:28 EDT 2018. Contains 316339 sequences. (Running on oeis4.)