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A003000
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Number of bifix-free (or primary, or unbordered) words of length n over a two-letter alphabet.
(Formerly M0328)
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36
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1, 2, 2, 4, 6, 12, 20, 40, 74, 148, 284, 568, 1116, 2232, 4424, 8848, 17622, 35244, 70340, 140680, 281076, 562152, 1123736, 2247472, 4493828, 8987656, 17973080, 35946160, 71887896, 143775792, 287542736, 575085472, 1150153322, 2300306644, 4600578044, 9201156088
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OFFSET
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0,2
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COMMENTS
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This is the number of binary words w of length n such that there is no nonempty word x, different from w, which is both a prefix and a suffix of w. - N. J. A. Sloane, Nov 09 2012
Many authors use the term "unbordered" for "bifix-free". The Lothaire (1997) reference refers to bifix-free words as primary words (Chapter 8). - David Callan, Sep 25 2006
Also the number of binary "prime palstars" of length 2n (Rampersad, Shallit, & Wang 2011). - Jeffrey Shallit, Aug 14 2014
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 28.
M. Lothaire, Combinatorics on Words, Cambridge University Press, NY, 1997, see p. 153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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L. J. Guibas and A. M. Odlyzko, Periods in Strings, Journal of Combinatorial Theory A 30 (1981) 19-42. Their L_n(0) is A003000(n).
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FORMULA
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a(2*n+1) = 2*a(2*n), a(2*n) = 2*a(2*n-1) - a(n).
G.f.: g(x) satisfies (1-2*x)*g(x) = 2 - g(x^2). - Robert Israel, Jan 12 2015
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EXAMPLE
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Bi-fix free words of lengths 1 through 4:
0, 1
10, 01
100, 110, 011, 001
1000, 1100, 1110, 0111, 0011, 0001.
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MAPLE
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A[0]:= 1:
for n from 1 to 100 do
if n::odd then A[n]:= 2*A[n-1] else A[n]:= 2*A[n-1]-A[n/2] fi
od:
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MATHEMATICA
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a[0]=1; a[n_]:=a[n]=2*a[n-1]-(1+(-1)^n)/2*a[Floor[n/2]]; Table[a[n], {n, 0, 34}]
a[0]=1; a[n_]:=a[n]=2*a[n-1]-If[EvenQ[n], a[n/2], 0] (* Ed Pegg Jr, Jan 05 2005 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Additional comments from Torsten.Sillke(AT)lhsystems.com, Jan 17 2001
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STATUS
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approved
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