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A102309 a(n) = Sum_{d divides n} moebius(d) * binomial(n/d,2).

%I A102309 #80 Jun 08 2025 05:01:27
%S A102309 0,0,1,3,5,10,11,21,22,33,34,55,46,78,69,92,92,136,105,171,140,186,
%T A102309 175,253,188,290,246,315,282,406,284,465,376,470,424,564,426,666,531,
%U A102309 660,568,820,570,903,710,852,781,1081,760,1155,890,1136,996,1378,963,1420,1140,1422,1246
%N A102309 a(n) = Sum_{d divides n} moebius(d) * binomial(n/d,2).
%C A102309 Zero followed by the Moebius transform of A000217. - _R. J. Mathar_, Jan 19 2009
%C A102309 Apparently, a(n-1) is the number of periodic complex Horadam orbits with period n, for n>2. - _Nathaniel Johnston_, Oct 04 2013
%C A102309 Also apparently, the first differences of A100448 (checked up to n=2000).
%H A102309 Alois P. Heinz, <a href="/A102309/b102309.txt">Table of n, a(n) for n = 0..10000</a>
%H A102309 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/978-3-030-51502-7">Recurrent Sequences: Key Results, Applications, and Problems</a>, Springer (2020), p. 159.
%H A102309 Ovidiu Bagdasar, <a href="/A102309/a102309.pdf">On certain computational and geometric properties of complex Horadam orbits</a>, poster, ANTS 2014.
%H A102309 Ovidiu D. Bagdasar and Peter J. Larcombe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/51-1/BagdasarLarcombe.pdf">On the characterization of periodic complex Horadam sequences</a>, Fib. Quart. 51 (1) (2013) 28-37.
%H A102309 Ovidiu D. Bagdasar and Peter J. Larcombe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/51-4/BagdasarLarcombe.pdf">On the Number of Complex Horadam Sequences with a Fixed Period</a>, Fib. Q., 51 (2013), 339-347.
%H A102309 Ovidiu D. Bagdasar and Peter J. Larcombe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-4/BagdasarLarcombe05252017.pdf">On the masked periodicity of Horadam sequences: a generator-based approach</a>, Fib. Q., 55 (2017), 332-339.
%H A102309 Ovidiu Bagdasar and Ioan-Lucian Popa, <a href="https://doi.org/10.1016/j.endm.2016.11.002">On the geometry of certain periodic non-homogeneous Horadam sequences</a>, Electronic Notes in Discrete Mathematics 56 (2016) 7-13.
%F A102309 G.f.: Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3. - _Seiichi Manyama_, May 24 2021
%F A102309 Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)). - _Amiram Eldar_, Jun 08 2025
%p A102309 with(numtheory):
%p A102309 a:= n-> add(mobius(d)*binomial(n/d, 2), d=divisors(n)):
%p A102309 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 18 2013
%t A102309 a[n_] := Sum[MoebiusMu[d] Binomial[n/d, 2], {d, Divisors[n]}];
%t A102309 a /@ Range[0, 60] (* _Jean-François Alcover_, Feb 04 2020 *)
%o A102309 (PARI) a(n) = sumdiv(n, d, moebius(d) * binomial(n/d,2) ); /* _Joerg Arndt_, Feb 18 2013 */
%o A102309 (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
%o A102309 (Python)
%o A102309 from math import comb
%o A102309 from sympy import mobius, divisors
%o A102309 def A102309(n): return sum(mobius(d)*comb(n//d,2) for d in divisors(n,generator=True)) # _Chai Wah Wu_, May 09 2025
%Y A102309 Second column of triangle A020921.
%Y A102309 Cf. A002117, A008683, A100448, A326419.
%K A102309 nonn
%O A102309 0,4
%A A102309 _Ralf Stephan_, Jan 03 2005