OFFSET
0,4
COMMENTS
Zero followed by the Moebius transform of A000217. - R. J. Mathar, Jan 19 2009
Apparently, a(n-1) is the number of periodic complex Horadam orbits with period n, for n>2. - Nathaniel Johnston, Oct 04 2013
Also apparently, the first differences of A100448 (checked up to n=2000).
The conjectured relation to A100448 is true (see Fried link). - Sela Fried, Mar 15 2026
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 159.
Ovidiu Bagdasar, On certain computational and geometric properties of complex Horadam orbits, poster, ANTS 2014.
Ovidiu D. Bagdasar and Peter J. Larcombe, On the characterization of periodic complex Horadam sequences, Fib. Quart. 51 (1) (2013) 28-37.
Ovidiu D. Bagdasar and Peter J. Larcombe, On the Number of Complex Horadam Sequences with a Fixed Period, Fib. Q., 51 (2013), 339-347.
Ovidiu D. Bagdasar and Peter J. Larcombe, On the masked periodicity of Horadam sequences: a generator-based approach, Fib. Q., 55 (2017), 332-339.
Ovidiu Bagdasar and Ioan-Lucian Popa, On the geometry of certain periodic non-homogeneous Horadam sequences, Electronic Notes in Discrete Mathematics 56 (2016) 7-13.
Sela Fried, Proof of a conjecture stated in A102309, 2026.
FORMULA
G.f.: Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3. - Seiichi Manyama, May 24 2021
Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)). - Amiram Eldar, Jun 08 2025
MAPLE
with(numtheory):
a:= n-> add(mobius(d)*binomial(n/d, 2), d=divisors(n)):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 18 2013
MATHEMATICA
a[n_] := Sum[MoebiusMu[d] Binomial[n/d, 2], {d, Divisors[n]}];
a /@ Range[0, 60] (* Jean-François Alcover, Feb 04 2020 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(d) * binomial(n/d, 2) ); /* Joerg Arndt, Feb 18 2013 */
(PARI) my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, moebius(k)*x^(2*k)/(1-x^k)^3))) \\ Seiichi Manyama, May 24 2021
(Python)
from math import comb
from sympy import mobius, divisors
def A102309(n): return sum(mobius(d)*comb(n//d, 2) for d in divisors(n, generator=True)) # Chai Wah Wu, May 09 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, Jan 03 2005
STATUS
approved
