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

Author

Thomas Ward, Mar 21 2001

Keywords

• nonn

Examples

			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.
		

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

Crossrefs

Cf. A008683, A047863.

Programs

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

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 *)

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

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

Formula

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

Extensions

More terms from Michel Marcus, Sep 11 2017