OFFSET
1,5
COMMENTS
Euler transform is A000045. 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)^2*(1-x^6)^2*...) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + ... - Michael Somos, Jan 28 2003
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..4000
Michael Baake, Joachim Hermisson, and Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.
Michael Baake, John A.G. Roberts, and Alfred Weiss, Periodic orbits of linear endomorphisms on the 2-torus and its lattices, arXiv:0808.3489 [math.DS], Aug 26, 2008. [Jonathan Vos Post, Aug 27 2008]
Larry Ericksen, Primality Testing and Prime Constellations, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.
N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, JIS 12 (2009) 09.4.5.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences
FORMULA
a(n) = (1/n)* Sum_{d|n} mu(d)*A001350(n/d).
a(n) = A006206(n) except for n=2. - Michael Somos, Jan 28 2003
a(n) = A031367(n)/n. - R. J. Mathar, Jul 15 2016
G.f.: Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k - x^(2*k)))/k. - Ilya Gutkovskiy, May 18 2019
EXAMPLE
a(7)=4 since the 7th term of A001350 is 29 and the 1st is 1, so there are (29-1)/7 = 4 orbits of length 7.
x + x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 11*x^10 + 18*x^11 + ...
MAPLE
A060280 := proc(n)
add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;
%/n;
end proc: # R. J. Mathar, Jul 15 2016
MATHEMATICA
A001350[n_] := LucasL[n] - (-1)^n - 1;
a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#]*A001350[n/#]& ];
Array[a, 50] (* Jean-François Alcover, Nov 23 2017 *)
PROG
(PARI) {a(n) = if( n<3, n==1, sumdiv( n, d, moebius(n/d) * (fibonacci(d - 1) + fibonacci(d + 1))) / n)} /* Michael Somos, Jan 28 2003 */
(Magma)
A060280:= func< n | n le 2 select 2-n else (&+[Lucas(d)*MoebiusMu(Floor(n/d)) : d in Divisors(n)])/n >;
[A060280(n): n in [1..50]]; // G. C. Greubel, Nov 06 2024
(SageMath)
A000032=BinaryRecurrenceSequence(1, 1, 2, 1)
def A060280(n): return sum(A000032(k)*moebius(n/k) for k in (1..n) if (k).divides(n))//n - int(n==2)
[A060280(n) for n in range(1, 41)] # G. C. Greubel, Nov 06 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Thomas Ward, Mar 29 2001
STATUS
approved