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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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+0
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.rs), 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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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
E. Deutsch and Lafayette College Problem Group, Problem 11161, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..300
R. Munafo, Enumeration of Period-N Mu-Atoms
Index entries for sequences related to Lyndon words
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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]
Sequence in context: A134774 A165729 A056278 this_sequence A161625 A069712 A076971
Adjacent sequences: A000737 A000738 A000739 this_sequence A000741 A000742 A000743
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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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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