A060280 Number of orbits of length n under the map whose periodic points are counted by A001350.

1, 0, 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, 22542396

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_] := Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n - 1;
a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#]*a001350[n/#]& ];
Array[a, 30] (* 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 */

Crossrefs

Cf. A000045, A001350, A006206, A324485, A324489. First column of A348422.

Sequence in context: A107458 A274142 A006206 * A095719 A153952 A050364

Adjacent sequences: A060277 A060278 A060279 * A060281 A060282 A060283

Keyword

easy,nonn

Author

Thomas Ward, Mar 29 2001

Status

approved