A102309 a(n) = Sum_{d divides n} moebius(d) * C(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

0,4

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).

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.

O. D. Bagdasar and P. J. Larcombe, On the characterization of periodic complex Horadam sequences, Fib. Quart. 51 (1) (2013) 28-37.

O. D. Bagdasar and P. 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 I.-L. Popa, On the geometry of certain periodic non-homogeneous Horadam sequences, Electronic Notes in Discrete Mathematics 56 (2016) 7-13.

Formula

G.f.: Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3. - Seiichi Manyama, May 24 2021

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

Crossrefs

Second column of triangle A020921.

Cf. A008683, A326419.

Sequence in context: A191513 A254540 A326419 * A067230 A321793 A075741

Adjacent sequences: A102306 A102307 A102308 * A102310 A102311 A102312

Keyword

nonn

Author

Ralf Stephan, Jan 03 2005

Status

approved