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Search: id:A000740
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| A000740 |
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Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.
(Formerly M2582 N1021)
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13
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| 1, 1, 1, 3, 6, 15, 27, 63, 120, 252, 495, 1023, 2010, 4095, 8127, 16365, 32640, 65535, 130788, 262143, 523770, 1048509, 2096127, 4194303, 8386440, 16777200, 33550335, 67108608, 134209530, 268435455, 536854005, 1073741823, 2147450880
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Also number of compositions of n into relatively prime parts. Also number of subsets of {1,2,..,n} containing n and consisting of relatively prime numbers. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 13 2003
Also number of perfect parity patterns that have exactly n columns (see A118141). - D. E. Knuth, May 11 2006
a(n) is odd if and only if n is squarefree (Tim Keller). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 27 2007
a(n) is a multiple of 3 for all n>=3 (see Problem 11161, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 13 2008]
Starting with offset 1, = Mobius transform (A054525) of [1, 2, 4, 8,...]; = row sums of triangle A143424. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 14 2008]
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REFERENCES
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H. W. Gould, Binomial coefficients, the bracket function, and compositions with relatively prime summands, Fib. Quart. 2 (1964), 241-260.
H. O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag; contribution by A. Douady, p. 165.
E. Deutsch and Lafayette College Problem Group, Problem 11161, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..300
More information
Index entries for sequences related to Lyndon words
R. Munafo, Enumeration of Period-N Mu-Atoms
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FORMULA
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2^{n-1} = sum {d|n} a(d); a(n) = Sum_{d|n} mu(n/d)*2^(d-1).
Rec. relation: a(n)=2^(n-1) - Sum(a(n/d), d|n, d>1) (Lafayette College Problem Group; see the Maple program). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 27 2007
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MAPLE
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with(numtheory): a[0]:=1: a[1]:=1: a[2]:=1: for n from 3 to 32 do div:=divisors(n): a[n]:=2^(n-1)-sum(a[n/div[j]], j=2..tau(n)) od: seq(a[n], n=0..32); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 27 2007
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PROGRAM
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(PARI) a(n)=if(n<1, n >= 0, sumdiv(n, d, moebius(n/d)*2^(d-1)))
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CROSSREFS
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Cf. A003239, A022553, A034738, A035928.
Cf. A000837.
A054525, A143424 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 14 2008]
Adjacent sequences: A000737 A000738 A000739 this_sequence A000741 A000742 A000743
Sequence in context: A079825 A134774 A056278 this_sequence A069712 A076971 A103529
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KEYWORD
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nonn,nice,easy
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AUTHOR
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njas
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EXTENSIONS
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Connection with Mandelbrot set discovered by Warren Smith (wds(AT)research.nj.nec.com), and proved by Robert Munafo (mrob(AT)mrob.com), Feb 06 2000
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