A102309 a(n) = Sum_{d divides n} moebius(d) * binomial(n/d,2).
0, 0, 1, 3, 5, 10, 11, 21, 22, 33, 34, 55, 46, 78, 69, 92, 92, 136, 105, 171, 140, 186, 175, 253, 188, 290, 246, 315, 282, 406, 284, 465, 376, 470, 424, 564, 426, 666, 531, 660, 568, 820, 570, 903, 710, 852, 781, 1081, 760, 1155, 890, 1136, 996, 1378, 963, 1420, 1140, 1422, 1246
Offset: 0
Keywords
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.
Programs
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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
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Mathematica
a[n_] := Sum[MoebiusMu[d] Binomial[n/d, 2], {d, Divisors[n]}]; a /@ Range[0, 60] (* Jean-François Alcover, Feb 04 2020 *) -
PARI
a(n) = sumdiv(n, d, moebius(d) * binomial(n/d,2) ); /* Joerg Arndt, Feb 18 2013 */
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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
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
Comments