A060224 Number of orbits of length n under the map whose periodic points are counted by A047863.

2, 2, 8, 39, 288, 3046, 47232, 1061100, 34385064, 1601137110, 106806380544, 10186152828755, 1386394018652160, 268976332493883474, 74301040560350828856, 29201332000320392849280, 16315436194909017151242240, 12952804290011844088808158188, 14603450579455204338154338779136

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..113

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

T. Ward, Exactly realizable sequences

Formula

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A047863(n/d).

Example

a(5)=288 since the 6th term of A047863 is 1442 and the 2nd term is 2, so there must be (1442-2)/5 = 288 orbits of length 5.

Mathematica

A047863[n_]:= A047863[n]= Sum[Binomial[n, k]*2^(k*(n-k)), {k, 0, n}];
A060224[n_]:= DivisorSum[n, MoebiusMu[#]*A047863[n/#] &]/n;
Table[A060224[n], {n, 40}] (* G. C. Greubel, Nov 03 2024 *)

Prog

(PARI) a047863(n) = n!*polcoeff(sum(k=0, n, exp(2^k*x +x*O(x^n))*x^k/k!), n);
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a047863(n/d)); \\ Michel Marcus, Sep 11 2017
(Magma)
A047863:= func< n | (&+[Binomial(n, k)*2^(k*(n-k)): k in [0..n]]) >;
A060224:= func< n | (&+[MoebiusMu(d)*A047863(Floor(n/d)): d in Divisors(n)])/n >;
[A060224(n): n in [1..40]]; // G. C. Greubel, Nov 03 2024
(SageMath)
def A047863(n): return sum(binomial(n, k)*2^(k*(n-k)) for k in range(n+1))
def A060224(n): return sum(moebius(k)*A047863(n//k) for k in (1..n) if (k).divides(n))//n
[A060224(n) for n in range(1, 41)] # G. C. Greubel, Nov 03 2024

Crossrefs

Cf. A008683, A047863.

Sequence in context: A009543 A102647 A318869 * A232980 A212307 A111605

Adjacent sequences: A060221 A060222 A060223 * A060225 A060226 A060227

Keyword

nonn

Author

Thomas Ward, Mar 21 2001

Extensions

More terms from Michel Marcus, Sep 11 2017

Status

approved