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

1,3

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.

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

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 *)

Crossrefs

Cf. A001350, A006206, A000032.

Sequence in context: A360620 A092434 A239632 * A073443 A257494 A302347

Adjacent sequences: A031364 A031365 A031366 * A031368 A031369 A031370

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers

Status

approved