0,5

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.

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).

The trace of a polynomial of degree n is the coefficient of x^(n-1); the subtrace is the coefficient of x^(n-2).

Also number of binary Lyndon words with trace 1 over GF(2).

Number of self-reciprocal polynomials of degree 2n over GF(2).

Contribution from Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009: (Start)

Also the number of dynamical cycles of period 2n of a threshold Boolean automata

network which is a quasi-minimal negative circuit of size nq where q is odd and

which is updated in parallel. (End)

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]

L. Carlitz, A theorem of Dickson on irreducible polynomials. Proc. Amer. Math. Soc. 3, (1952). 693-700.

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]

N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.

R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.

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.

R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200

Joerg Arndt, Fxtbook

H. Meyn and W. G\"otz, Self-reciprocal polynomials over finite fields

F. Ruskey, Number of q-ary Lyndon words with given trace mod q

F. Ruskey, Number of Lyndon words of given trace

N. J. A. Sloane, On single-deletion-correcting codes

J.-Y. Thibon, The cycle enumerator of unimodal permutations.

Index entries for "core" sequences

Index entries for sequences related to Lyndon words

Index entries for sequences related to subset sums modulo m

(Sum_{odd d divides n } mu(d)*2^(n/d)) / (2n).

a(5) = 3 corresponding to the necklaces 00001, 00111, 01011; a(6) = 5 from 000001, 000011, 000101, 000111, 001011.

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;

(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]

Like A000013, but primitive necklaces. Half of A064355.

Equals A042981 + A042982. Cf. A002823, A000016, A053633, A051841, A001037, A002075, A002076.

Cf. A001037 [From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009]

Sequence in context: A143961 A128023 A056303 this_sequence A074099 A006788 A054650

Adjacent sequences: A000045 A000046 A000047 this_sequence A000049 A000050 A000051

nonn,core,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Frank Ruskey (fruskey(AT)cs.uvic.ca), Dec 13 1999