

A005434


Number of distinct autocorrelations of binary words of length n.
(Formerly M0555)


8



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

1,2


COMMENTS

Conjecture: a(n + 1)  a(n) < a(n + 13)  a(n + 12) for all n >= 1.  Eric Rowland, Nov 24 2021


REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, AddisonWesley 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).


LINKS

EHern Lee, Table of n, a(n) for n = 1..654
L. J. Guibas, Periodicities in Strings, Combinatorial Algorithms on Words 1985, NATO ASI Vol. F12, 257269
L. J. Guibas and A. M. Odlyzko, Periods in Strings, Journal of Combinatorial Theory A 30:1 (1981) 1942.
Leo J. Guibas and Andrew M. Odlyzko, String overlaps, pattern matching and nontransitive games, Journal of Combinatorial Theory Series A, 30 (March 1981), 183208.
H. Harborth, Endliche 01Folgen mit gleichen Teilblöcken, Journal für Mathematik, 271 (1974) 139154.
A. Kaseorg, Rust program used to compute values for n up to 500
E. H. Rivals and S. Rahmann, Combinatorics of Periods in Strings, Journal of Combinatorial Theory  Series A, Vol. 104(1) (2003), pp. 95113.
E. H. Rivals, Autocorrelation of Strings.
E. H. Rivals and S. Rahmann Combinatorics of Periods in Strings
T. Sillke, Autocorrelation Range
T. Sillke, kappa sequence for words of length n
T. Sillke, The autocorrelation function


EXAMPLE

From Eric Rowland, Nov 22 2021: (Start)
For n = 5 there are a(5) = 6 distinct autocorrelations of length5 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)


MAPLE

A005434 := proc( n :: posint )
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 )
end proc: # James McCarron, Jun 21 2017


MATHEMATICA

Table[Length[Union[Map[Flatten[Position[Table[Take[#, ni]==Drop[#, i], {i, 0, n1}], True]1]&, Tuples[{0, 1}, n]]]], {n, 1, 15}] (* Geoffrey Critzer, Nov 29 2013 *)


CROSSREFS

Cf. A018819 (related to a lower bound for autocorrelations), A045690 (the number of binary strings sharing the same autocorrelation).
Sequence in context: A325350 A027585 A123015 * A027589 A039851 A239100
Adjacent sequences: A005431 A005432 A005433 * A005435 A005436 A005437


KEYWORD

nonn,nice


AUTHOR

Simon Plouffe, N. J. A. Sloane


EXTENSIONS

More terms and additional references from torsten.sillke(at)lhsystems.com
Definition clarified by Eric Rowland, Nov 22 2021


STATUS

approved



