%I A000048 M0711 N0262
%S A000048 1,1,1,1,2,3,5,9,16,28,51,93,170,315,585,1091,2048,3855,7280,13797,
%T A000048 26214,49929,95325,182361,349520,671088,1290555,2485504,4793490,
%U A000048 9256395,17895679,34636833,67108864,130150493,252645135,490853403
%N A000048 Number of n-bead necklaces with beads of 2 colors and primitive period
n, when turning over is not allowed but the two colors can be interchanged.
%C A000048 Also 2n-bead balanced binary necklaces of fundamental period 2n that
are equivalent to their complements; binary Lyndon words of length
n with an odd number of 1's; number of binary irreducible polynomials
of degree n having trace 1.
%C A000048 Also number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i
= 1 (mod n+1) = size of Varshamov-Tenengolts code VT_1(n).
%C A000048 The trace of a polynomial of degree n is the coefficient of x^(n-1);
the subtrace is the coefficient of x^(n-2).
%C A000048 Also number of binary Lyndon words with trace 1 over GF(2).
%C A000048 Number of self-reciprocal polynomials of degree 2n over GF(2).
%C A000048 Contribution from Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar
03 2009: (Start)
%C A000048 Also the number of dynamical cycles of period 2n of a threshold Boolean
automata
%C A000048 network which is a quasi-minimal negative circuit of size nq where q
is odd and
%C A000048 which is updated in parallel. (End)
%C A000048 Also the number of 3-elements orbits of the symmetric group S3 action
on irreducible polynomials of degree 2n, n>1, over GF(2). [From Jean-Francis
Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Oct
04 2009]
%D A000048 L. Carlitz, A theorem of Dickson on irreducible polynomials. Proc. Amer.
Math. Soc. 3, (1952). 693-700.
%D A000048 J. Demongeot, M. Noual and S. Sene, On the number of attractors of positive
and negative threshold Boolean automata circuits", preprint (2009)
[From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009]
%D A000048 N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958),
285-302.
%D A000048 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois
J. Math., 5 (1961), 657-665.
%D A000048 B. D. Ginsburg, On a number theory function applicable in coding theory,
Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
%D A000048 R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer.
Math. Monthly, 113 (2006), 887-896.
%D A000048 H. Kawakami, Table of rotation sequences of x_{n+1} = x_n^2 - lambda,
pp. 73-92 of G. Ikegami, Editor, Dynamical Systems and Nonlinear
Oscillations, Vol. 1, World Scientific, 1986.
%D A000048 R. M. May, Simple mathematical models with very complicated dynamics,
Nature, 261 (Jun 10, 1976), 459-467.
%D A000048 N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for
transformations on the unit interval, J. Combin. Theory, A 15 (1973),
25-44; reprinted in P. Cvitanovic, ed., Universality in Chaos, Hilger,
Bristol, 1986, pp. 187-206.
%D A000048 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000048 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs
(Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst.
Publ., 10, de Gruyter, Berlin, 2002.
%D A000048 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A000048 T. D. Noe, <a href="b000048.txt">Table of n, a(n) for n = 0..200</a>
%H A000048 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000048 H. Meyn and W. G\"otz, <a href="http://www.mat.univie.ac.at/~slc/opapers/
s21meyn.html">Self-reciprocal polynomials over finite fields</a>
%H A000048 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/neck/lyndon.html">
Number of q-ary Lyndon words with given trace mod q</a>
%H A000048 F. Ruskey, <a href="http://www.theory.csc.uvic.ca/~cos/inf/trs/lyn/Fq/
lyn_tr_Fq.html">Number of Lyndon words of given trace</a>
%H A000048 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/dijen.txt">
On single-deletion-correcting codes</a>
%H A000048 J.-Y. Thibon, <a href="http://www.arXiv.org/abs/math.CO/0102051">The
cycle enumerator of unimodal permutations</a>.
%H A000048 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000048 <a href="Sindx_Lu.html#Lyndon">Index entries for sequences related to
Lyndon words</a>
%H A000048 <a href="Sindx_Su.html#subsetsums">Index entries for sequences related
to subset sums modulo m</a>
%F A000048 (Sum_{odd d divides n } mu(d)*2^(n/d)) / (2n).
%e A000048 a(5) = 3 corresponding to the necklaces 00001, 00111, 01011; a(6) = 5
from 000001, 000011, 000101, 000111, 001011.
%p A000048 with(numtheory); A000048 := proc(n) local d,t1; if n = 0 then RETURN(1)
else t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1
then t1 := t1+mobius(d)*2^(n/d)/(2*n); fi; od; RETURN(t1); fi; end;
%o A000048 (PARI) A000048(n) = sumdiv(n,d,(d%2)*(moebius(d)*2^(n/d)))/(2*n) [From
Michael Porter (michael_b_porter(AT)yahoo.com), Nov 09 2009]
%Y A000048 Like A000013, but primitive necklaces. Half of A064355.
%Y A000048 Equals A042981 + A042982. Cf. A002823, A000016, A053633, A051841, A001037,
A002075, A002076.
%Y A000048 Cf. A001037 [From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar
03 2009]
%Y A000048 Sequence in context: A143961 A128023 A056303 this_sequence A074099 A006788
A054650
%Y A000048 Adjacent sequences: A000045 A000046 A000047 this_sequence A000049 A000050
A000051
%K A000048 nonn,core,easy,nice,new
%O A000048 0,5
%A A000048 N. J. A. Sloane (njas(AT)research.att.com).
%E A000048 Additional comments from Frank Ruskey (fruskey(AT)cs.uvic.ca), Dec 13
1999
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