1,5

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+...

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.

Related to Fibonacci sequence since 1/(1 - x^n - x^(2n)) expands to a power series whose terms are Fibonacci numbers.

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

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

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. 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

B.-S. Du, A simple method which generates infinitely many congruence identities, Fib. Quart., 27 (1989), 116-124.

Joerg Arndt, Fxtbook

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 UTA-PHYS-96-44, hep-th/9609128

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

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

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.

(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).

Round ((1/n) sum_{ d divides n } mu(n) phi^(n/d)) (formula from D.Broadhurst(AT)open.ac.uk).

G.f.: Sum_{n=1..inf} -mu(n)*ln(1 - x^n - x^(2n))/n.

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]

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

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;

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

Equals A031367(n)/n. Equals A060280 except for n=2.

Sequence in context: A013979 A107458 A060280 this_sequence A095719 A153952 A050364

Adjacent sequences: A006203 A006204 A006205 this_sequence A006207 A006208 A006209

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com) and Frank Ruskey (fruskey(AT)cs.uvic.ca)

Replaced arXiv URL by non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2009