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.

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, 954444608, 1857283156, 3616828364

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• changed

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

Examples

			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 + ...
		

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). See also arXiv:2308.16257 [math.CO], 2023. See references.
• Joerg Arndt, Matters Computational (The Fxtbook), p. 151, p. 408.
• Raghav Bhutani and Frederick Saia, Replacement dynamics of binary quadratic forms, arXiv:2508.05816 [math.NT], 2025. See p. 6.
• Henry Bottomley, Initial terms of A000011 and A000013
• Zachary E. Chin and Isaac L. Chuang, The quantum trajectory sensing problem and its solution, arXiv:2410.00893 [quant-ph], 2024. See p. 19.
• 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.
• Darij Grinberg and Peter Mao, Necklaces over a group with identity product, arXiv:2405.08937 [math.CO], 2024. See pp. 15, 22.
• 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

Crossrefs

Cf. A000031, A000016, A000116.

Cf. A128976.

Cf. A000010, A027750.

Cf. A385665, A000048.

Programs

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

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 *)

PARI

{a(n) = if( n<1, n==0, sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n))}; /* Michael Somos, Oct 20 1999 */

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

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

a(n) = Sum_{k=1..n} A385665(n,k) = Sum_{d|n} A000048(d). - Tilman Piesk, Jul 31 2025