A000013 Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed. (Formerly M0313 N0115)

1, 1, 2, 2, 4, 4, 8, 10, 20, 30, 56, 94, 180, 316, 596, 1096, 2068, 3856, 7316, 13798, 26272, 49940, 95420, 182362, 349716, 671092, 1290872, 2485534, 4794088, 9256396, 17896832, 34636834, 67110932, 130150588, 252648992, 490853416

Offset

0,3

Comments

Definition (2): Equivalently, number of different output sequences from an n-stage pure cycling shift register when 2 sequences are considered the same if one is the complement of the other.

Definition (3): Also number of different output sequences from an n-stage pure cycling shift register constrained so contents have even weight.

Definition (4): Also number of output sequences from (n-1)-stage shift register which feeds back the mod 2 sum of the contents of the register.

The equivalence of definitions (1) and (2) follows at once from the definitions.

If u is an output sequence of type (2) then its derivative is of type (3) - so (2) and (3) count the same things.

If we have a shift register of type (4), append a new cell which contains the mod 2 sum of the contents to get a shift register of type (3). So (3) and (4) count the same things.

If n is even, a(n) = A000116(n/2). If 2^(n+1)-1 is prime, then a(n) = A128976(n+1), the number of cycles in the digraph of the Lucas-Lehmer operator LL(x) = x^2 - 2 acting on Z/(2^(n+1)-1). - M. F. Hasler, May 19 2007

Also number of 2n-bead balanced binary necklaces that are equivalent to their complements. - Andrew Howroyd, Sep 29 2017

References

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)

Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023).

Joerg Arndt, Matters Computational (The Fxtbook), p. 151, p. 408.

Henry Bottomley, Initial terms of A000011 and A000013

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.

Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.

Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

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

N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; 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, Maple code for this and related sequences

Index entries for sequences related to necklaces

Formula

a(n) = Sum_{ d divides n } (phi(2*d)*2^(n/d))/(2*n) for n>0. - Michael Somos, Oct 20 1999

G.f.: 1 - Sum_{i>=1} phi(2*i)*log(1-2*x^i)/(2*i). - Herbert Kociemba, Nov 01 2016

From Richard L. Ollerton, May 11 2021: (Start)

For n >= 1:

a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.

a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)

a(n) ~ 2^(n-1)/n. - Cedric Lorand, Apr 24 2022

Example

G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 10*x^7 + 20*x^8 + ...

Maple

with(numtheory): A000013 := proc(n) local s, d; if n = 0 then RETURN(1) else s := 0; for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s); fi; end;

Mathematica

a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 0, Divisors[n]]
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (2 n)]; (* Michael Somos, Dec 19 2014 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[2i] Log[1-2*x^i]/(2i), {i, 1, mx}], {x, 0, mx}], x] (* Herbert Kociemba, Nov 01 2016 *)

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n))}; /* Michael Somos, Oct 20 1999 */
(Haskell)
a000013 0 = 1
a000013 n = sum (zipWith (*)
   (map (a000010 . (* 2)) ds) (map (2 ^) $ reverse ds)) `div` (2 * n)
   where ds = a027750_row n
-- Reinhard Zumkeller, Jul 08 2013
(Python)
from sympy import divisors, totient
def a(n): return 1 if n<1 else sum([totient(2*d)*2**(n/d) for d in divisors(n)])/(2*n) # Indranil Ghosh, Apr 28 2017

Crossrefs

Cf. A000031, A000016, A000116.

Cf. A128976.

Cf. A000010, A027750.

Sequence in context: A022476 A347206 A307240 * A064484 A063776 A287135

Adjacent sequences: A000010 A000011 A000012 * A000014 A000015 A000016

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved