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A006206
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Number of aperiodic binary necklaces of length n with no subsequence 00, excluding the necklace "0".
(Formerly M0317)
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22
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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
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 499.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
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, arXiv:hep-th/9609128, 1996.
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, arXiv:0706.2297 [math.DS], 2007; Bull. Austral. Math. Soc. 31 (1985), 89-103. Corrigendum: 32 (1985), 159.
Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, arXiv:0706.2421 [math.NT], 2007; Fib. Quart. 27 (1989), 116-124.
Larry Ericksen, Primality Testing and Prime Constellations, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, 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
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FORMULA
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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.
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EXAMPLE
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Necklaces are: 1; 10; 110; 1110; 11110, 11010; 111110, 111010; ...
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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: A013979 A107458 A274142 * A060280 A095719 A153952
Adjacent sequences: A006203 A006204 A006205 * A006207 A006208 A006209
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane and Frank Ruskey
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STATUS
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approved
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