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A133267
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Number of Lyndon words on {1, 2, 3} with an even number of 1's.
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3
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2, 1, 4, 8, 24, 56, 156, 400, 1092, 2928, 8052, 22080, 61320, 170664, 478288, 1344800, 3798240, 10760568, 30585828, 87166656, 249055976, 713197848, 2046590844, 5883926400, 16945772184, 48881973840, 141214767876, 408513980160
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| A Lyndon word is the aperiodic necklace representative which is lexicographically smallest among its cyclic shifts. Thus we can apply the fixed density formulas: L_k(n,d)=sum L(n-d, n_1,..., n_(k-1)); n_1+...+n_(k-1)=d where L(n_0, n_1,...,n_(k-1))=(1/n)sum mu(j)*[(n/j)!/((n_0/j)!(n_1/j)!...(n_(k-1)/j)!)]; j|gcd(n_0, n_1,...,n_(k-1)). For this sequence, sum over n_0=even. Alternatively, a(n)=(sum mu(d)*3^(n/d)/n; d|n) - (sum mu(d)*(3^(n/d)-1)/(2n); d|n, d odd).
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REFERENCES
| E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983.
F. Ruskey and J. Sawada, "An Efficient Algorithm for Generating Necklaces with Fixed Density", SIAM J. Computing, 29 (1999) 671-684.
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LINKS
| F. Ruskey, Counting Necklaces
M. Zabrocki, Course website
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FORMULA
| a(1)=2; for n>1, if n=2^k for some k, then a(n)=((3^(n/2)-1)^2)/(2n). Otherwise, if n=even then a(n)=sum mu(d)*(3^(n/d)-2*3^(n/(2d))/(2n); d|n, d odd. If n=odd then a(n)=sum mu(d)*(3^(n/d)-1)/(2n); d|n, d odd.
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EXAMPLE
| For n=3, out of 8 possible Lyndon words: 112, 113, 122, 123, 132, 133, 223, 233, only the first two and the last two have an even number of 1's. Thus a(3) = 4.
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MAPLE
| with (numtheory): a:= n-> add (mobius(d) *3^(n/d), d=divisors(n))/n -add (mobius(d) *(3^(n/d)-1), d=select (x-> irem(x, 2)=1, divisors(n)))/(2*n): seq (a(n), n=1..30); # Alois P. Heinz, Jul 29 2011
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CROSSREFS
| Cf. A006575, A027376.
Sequence in context: A156817 A008301 A113820 * A145864 A182739 A076014
Adjacent sequences: A133264 A133265 A133266 * A133268 A133269 A133270
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KEYWORD
| nonn
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AUTHOR
| Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Jan 03 2008
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