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A005434
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Number of distinct autocorrelations of binary words of length n.
(Formerly M0555)
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8
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1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 27, 30, 37, 47, 57, 62, 75, 87, 102, 116, 135, 155, 180, 194, 220, 254, 289, 312, 359, 392, 438, 479, 538, 595, 664, 701, 772, 863, 956, 1005, 1115, 1205, 1317, 1414, 1552, 1677, 1836, 1920, 2074, 2249, 2444
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n + 1) - a(n) < a(n + 13) - a(n + 12) for all n >= 1. - Eric Rowland, Nov 24 2021
log(a(n))/log^2(n) converges when n tends to infinity. This conjecture was first stated in (Guibas and Odlyzko, JCTA, 1981a). (Rivals et al. ICALP 2023) proves this conjecture and provides an improved upper bound for this ratio.
An autocorrelation is a binary encoding of the period set.
This sequence is also the number of autocorrelation for words over any finite alphabet whose cardinality is at least two. The autocorrelation is independent of the alphabet cardinality, provided the cardinality is at least two; see proofs in (Guibas and Odlyzko, JCTA, 1981a). (End)
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ., 2nd Ed., 1994. Section 8.4: Flipping Coins
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 30:1 (1981) 19-42.
E. Rivals, and S. Rahmann (2001). Combinatorics of Periods in Strings. In: Orejas, F., Spirakis, P.G., van Leeuwen, J.(eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. doi:10.1007/3-540-48224-5_51.
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EXAMPLE
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For n = 5 there are a(5) = 6 distinct autocorrelations of length-5 binary words:
00000 can overlap itself in 1, 2, 3, 4, or 5 letters. Its autocorrelation is 11111.
00100 can overlap itself in 1, 2, or 5 letters. Its autocorrelation is 10011.
01010 can overlap itself in 1, 3, or 5 letters. Its autocorrelation is 10101.
00010 can overlap itself in 1 or 5 letters. Its autocorrelation is 10001.
01001 can overlap itself in 2 or 5 letters. Its autocorrelation is 10010.
00001 can only overlap itself in 5 letters. Its autocorrelation is 10000.
(End)
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MAPLE
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local S := table();
for local c in Iterator:-BinaryGrayCode( n ) do
c := convert( c, 'list' );
S[ [seq]( evalb( c[ 1 .. i + 1 ] = c[ n - i .. n ] ), i = 0 .. n - 1 ) ] := 0
end do;
numelems( S )
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MATHEMATICA
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Table[Length[Union[Map[Flatten[Position[Table[Take[#, n-i]==Drop[#, i], {i, 0, n-1}], True]-1]&, Tuples[{0, 1}, n]]]], {n, 1, 15}] (* Geoffrey Critzer, Nov 29 2013 *)
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CROSSREFS
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Cf. A018819 (related to a lower bound for autocorrelations), A045690 (the number of binary strings sharing the same autocorrelation).
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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More terms and additional references from torsten.sillke(at)lhsystems.com
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STATUS
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approved
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