login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006206 Number of aperiodic binary necklaces of length n with no subsequence 00, excluding the necklace "0".
(Formerly M0317)
21
1, 1, 1, 1, 2, 2, 4, 5, 8, 11, 18, 25, 40, 58, 90, 135, 210, 316, 492, 750, 1164, 1791, 2786, 4305, 6710, 10420, 16264, 25350, 39650, 61967, 97108, 152145, 238818, 374955, 589520, 927200, 1459960, 2299854, 3626200, 5720274, 9030450, 14263078 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Bau-Sen Du (1985/1989)'s Table 1 has this sequence, denoted A_{n,1}, as the second column. - Jonathan Vos Post, Jun 18 2007

REFERENCES

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

LINKS

James Spahlinger, Table of n, a(n) for n = 1..5000

Joerg Arndt, Matters Computational (The Fxtbook), p.710

Michael Baake, Joachim Hermisson, Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.

D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett B., 393 (1997), p 403. UTA-PHYS-96-44, arXiv:hep-th/9609128

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Marston Conder, S. Du, R. Nedela, M. Skoviera, Regular maps with nilpotent automorphism group, Journal of Algebraic Combinatorics, December 2016, Volume 44, Issue 4, pp 863-874; DOI: 10.1007/s10801-016-0692-8. ["... We note that the sequence h_n above agrees in all but the first term with the sequence A006206 in ..."]

Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem. Bull. Austral. Math. Soc. 31 (1985), 89-103. Corrigendum: 32 (1985), 159.

Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart. 27 (1989), 116-124.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, sequence gamma_{1,j}^(A).

A. Pakapongpun, T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 21.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Index entries for sequences related to Lyndon words

FORMULA

Euler transform is Fibonacci(n+1): 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)^2*(1-x^6)^2...)= 1/prod(n>=1, (1-x^n)^a(n) ) = 1+x+2*x^2+3*x^3+5*x^4+8*x^5+...

Coefficients of power series of natural logarithm of the infinite product prod(n>=1, (1 - x^n - x^(2*n))^(-mu(n)/n) ), where mu(n) is the Moebius function. This is related to Fibonacci sequence since 1/(1 - x^n - x^(2*n)) expands to a power series whose terms are Fibonacci numbers.

a(n) = (1/n) * sum_{d|n} mu(n/d) * (Fib(d-1)+Fib(d+1)) = (1/n) * sum_{d|n} mu(n/d) * Lucas(d). Hence Lucas(n) = sum_{d|n} d*a(d).

a(n) = round((1/n)*sum_{d|n} mu(n)*phi^(n/d)) ). - David Broadhurst.

G.f.: sum(n>=1, -mu(n)*log(1 - x^n - x^(2*n))/n ).

a(n) = (1/n) * sum_{d|n} mu(d)*A001610(n/d-1), n>1. - R. J. Mathar, Mar 07 2009

For n>2, a(n) = A060280(n) = A031367(n)/n.

EXAMPLE

Necklaces are: 1; 10; 110; 1110; 11110, 11010; 111110, 111010; ...

MAPLE

A006206 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + mobius(n/d)*(fibonacci(d+1)+fibonacci(d-1)) end do; sum/n; end proc:

MATHEMATICA

a[n_] := Total[(MoebiusMu[n/#]*(Fibonacci[#+1] + Fibonacci[#-1]) & ) /@ Divisors[n]]/n;

(* or *) a[n_] := Sum[LucasL[k]*MoebiusMu[n/k], {k, Divisors[n]}]/n; Table[a[n], {n, 100}] (* Jean-François Alcover, Jul 19 2011, after given formulas *)

PROG

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/n)

(Haskell)

a006206 n = sum (map f $ a027750_row n) `div` n where

   f d = a008683 (n `div` d) * (a000045 (d - 1) + a000045 (d + 1))

-- Reinhard Zumkeller, Jun 01 2013

CROSSREFS

Cf. A006207 (A_{n,2}), A006208 (A_{n,3}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

Cf. A001461 (partial sums); A000045, A008683, A027750.

Sequence in context: A107458 A274142 A060280 * A095719 A153952 A050364

Adjacent sequences:  A006203 A006204 A006205 * A006207 A006208 A006209

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane and Frank Ruskey

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 25 05:56 EDT 2017. Contains 287012 sequences.