A060280 - OEIS

%I #52 Nov 06 2024 04:30:13

%S 1,0,1,1,2,2,4,5,8,11,18,25,40,58,90,135,210,316,492,750,1164,1791,

%T 2786,4305,6710,10420,16264,25350,39650,61967,97108,152145,238818,

%U 374955,589520,927200,1459960,2299854,3626200,5720274,9030450,14263078,22542396

%N Number of orbits of length n under the map whose periodic points are counted by A001350.

%C 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

%H Seiichi Manyama, <a href="/A060280/b060280.txt">Table of n, a(n) for n = 1..4000</a>

%H Michael Baake, Joachim Hermisson, and Peter Pleasants, <a href="http://dx.doi.org/10.1088/0305-4470/30/9/016">The torus parametrization of quasiperiodic LI-classes</a> J. Phys. A 30 (1997), no. 9, 3029-3056.

%H Michael Baake, John A.G. Roberts, and Alfred Weiss, <a href="http://arxiv.org/abs/0808.3489">Periodic orbits of linear endomorphisms on the 2-torus and its lattices</a>, arXiv:0808.3489 [math.DS], Aug 26, 2008. [_Jonathan Vos Post_, Aug 27 2008]

%H Larry Ericksen, <a href="http://siauliaims.su.lt/index.php?option=com_content&amp;view=article&amp;id=44&amp;Itemid=9">Primality Testing and Prime Constellations</a>, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.

%H N. Neumarker, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/">Realizability of Integer Sequences as Differences of Fixed Point Count Sequences</a>, JIS 12 (2009) 09.4.5.

%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>

%F a(n) = (1/n)* Sum_{d|n} mu(d)*A001350(n/d).

%F a(n) = A006206(n) except for n=2. - _Michael Somos_, Jan 28 2003

%F a(n) = A031367(n)/n. - _R. J. Mathar_, Jul 15 2016

%F G.f.: Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k - x^(2*k)))/k. - _Ilya Gutkovskiy_, May 18 2019

%e 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.

%e 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 + ...

%p A060280 := proc(n)

%p     add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;

%p     %/n;

%p end proc: # _R. J. Mathar_, Jul 15 2016

%t A001350[n_] := LucasL[n] - (-1)^n - 1;

%t a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#]*A001350[n/#]& ];

%t Array[a, 50] (* _Jean-François Alcover_, Nov 23 2017 *)

%o (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 */

%o (Magma)

%o A060280:= func< n | n le 2 select 2-n else (&+[Lucas(d)*MoebiusMu(Floor(n/d)) : d in Divisors(n)])/n >;

%o [A060280(n): n in [1..50]]; // _G. C. Greubel_, Nov 06 2024

%o (SageMath)

%o A000032=BinaryRecurrenceSequence(1,1,2,1)

%o def A060280(n): return sum(A000032(k)*moebius(n/k) for k in (1..n) if (k).divides(n))//n - int(n==2)

%o [A060280(n) for n in range(1,41)] # _G. C. Greubel_, Nov 06 2024

%Y Cf. A000032, A000045, A001350, A006206, A324485, A324489.

%Y First column of A348422.

%K easy,nonn

%O 1,5

%A _Thomas Ward_, Mar 29 2001