Search: id:A006206
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%I A006206 M0317
%S A006206 1,1,1,1,2,2,4,5,8,11,18,25,40,58,90,135,210,316,492,750,1164,1791,2786,
%T A006206 4305,6710,10420,16264,25350,39650,61967,97108,152145,238818,374955,
%U A006206 589520,927200,1459960,2299854,3626200,5720274,9030450,14263078
%N A006206 Number of aperiodic binary necklaces of length n with no subsequence 
               00, excluding the necklace "0".
%C A006206 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+x+2x^2+3x^3+5x^4+8x^\
               5+...
%C A006206 Coefficients of power series of natural logarithm of the infinite product 
               Product_{n=1..inf} (1 - x^n - x^(2n))^(-mu(n)/n), where mu(n) is 
               the Moebius function.
%C A006206 Related to Fibonacci sequence since 1/(1 - x^n - x^(2n)) expands to a 
               power series whose terms are Fibonacci numbers.
%C A006206 Bau-Sen Du (1985)'s Table 1, p. 6, has this sequence as the second column. 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 18 2007
%D A006206 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006206 Baake, Michael; Hermisson, Joachim; Pleasants, Peter A. B.; The torus 
               parametrization of quasiperiodic LI-classes. J. Phys. A 30 (1997), 
               no. 9, 3029-3056.
%D A006206 D. J. Broadhurst and D. Kreimer, Association of multiple zeta values 
               with positive knots via Feynman diagrams up to 9 loops, Phys. Lett 
               B. vol. 393 (1997) p 403
%D A006206 B.-S. Du, A simple method which generates infinitely many congruence 
               identities, Fib. Quart., 27 (1989), 116-124.
%H A006206 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A006206 D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the 
               enumeration of irreducible k-fold Euler sums and their roles in knot 
               theory and field theory</a>
%H A006206 D. J. Broadhurst and D. Kreimer, <a href="http://arXiv.org/abs/hep-th/
               9609128">Association of multiple zeta values with positive knots 
               via Feynman diagrams up to 9 loops</a> UTA-PHYS-96-44, hep-th/9609128
%H A006206 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A006206 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A006206 <a href="Sindx_Lu.html#Lyndon">Index entries for sequences related to 
               Lyndon words</a>
%H A006206 Bau-Sen Du, <a href="http://arXiv.org/abs/0706.2297">The Minimal Number 
               of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem</
               a>. Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 
               159.
%F A006206 (1/n)* sum_{ d divides n } mu(n/d) [ Fib(d-1)+Fib(d+1) ]; or (1/n) * 
               sum over d divides n of {mu(n/d) * Lucas_d}. Hence Lucas(n) = sum 
               over d divides n of d*a(d).
%F A006206 Round ((1/n) sum_{ d divides n } mu(n) phi^(n/d)) (formula from D.Broadhurst(AT)open.ac.uk).
%F A006206 G.f.: Sum_{n=1..inf} -mu(n)*ln(1 - x^n - x^(2n))/n.
%F A006206 n*a(n)= sum_{d|n} mu(d)*A001610(n/d-1), n>1. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 07 2009]
%e A006206 Necklaces are: 1; 10; 110; 1110; 11110, 11010; 111110, 111010; ...
%p A006206 A006206 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := 
               sum + mobius(n/d)*(fibonacci(d+1)+fibonacci(d-1)) od; RETURN(sum/
               n); end;
%o A006206 (PARI) a(n)=if(n<1,0,sumdiv(n,d,moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/
               n)
%Y A006206 Equals A031367(n)/n. Equals A060280 except for n=2.
%Y A006206 Sequence in context: A013979 A107458 A060280 this_sequence A095719 A153952 
               A050364
%Y A006206 Adjacent sequences: A006203 A006204 A006205 this_sequence A006207 A006208 
               A006209
%K A006206 nonn,easy,nice
%O A006206 1,5
%A A006206 N. J. A. Sloane (njas(AT)research.att.com) and Frank Ruskey (fruskey(AT)cs.uvic.ca)
%E A006206 Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 07 2009

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