A031367 Inflation orbit counts.

1, 0, 3, 4, 10, 12, 28, 40, 72, 110, 198, 300, 520, 812, 1350, 2160, 3570, 5688, 9348, 15000, 24444, 39402, 64078, 103320, 167750, 270920, 439128, 709800, 1149850, 1859010, 3010348, 4868640, 7880994, 12748470, 20633200, 33379200, 54018520, 87394452, 141421800

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also number of primitive Lucas strings of length n [Ashrafi et al.] - N. J. A. Sloane, Nov 19 2014

The preceding comment is true for all n except n=2, as there are 2 primitive Lucas strings of length 2. The sequence of the number of primitive Lucas strings is the Möbius transform of the Lucas numbers A000032. - Pontus von Brömssen, Jan 24 2019

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2000
• A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
• A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 3.
• Michael Baake, Joachim Hermisson, Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.

Crossrefs

Cf. A001350, A006206, A000032.

Programs

Maple

A031367 := proc(n)
    add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Jul 15 2016
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)/i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) a(n):= ((<<0|1>, <1|1>>^n)[1, 2]-b(n, n-1))*n end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 22 2018

Mathematica

a[n_] := n*Sum[MoebiusMu[d]*Sum[Binomial[k-1, 2k-n/d]/(n-d*k), {k, 0, n/d-1} ], {d, Divisors[n]}];
Array[a, 40] (* Jean-François Alcover, Jul 09 2018 *)

Formula

If b(n) is the n-th term of A001350, then a(n) = Sum_{d|n} mu(d)b(n/d).

a(n) = n * A060280(n).

G.f.: Sum_{k>=1} mu(k) * x^k * (1 + x^(2*k)) / ((1 - x^(2*k)) * (1 - x^k - x^(2*k))). - Ilya Gutkovskiy, Feb 06 2020

Extensions

More terms from James Sellers