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%I #68 Jan 05 2025 19:51:38
%S 0,0,1,3,5,10,11,21,22,33,34,55,46,78,69,92,92,136,105,171,140,186,
%T 175,253,188,290,246,315,282,406,284,465,376,470,424,564,426,666,531,
%U 660,568,820,570,903,710,852,781,1081,760,1155,890,1136,996,1378,963,1420,1140,1422,1246
%N a(n) = Sum_{d divides n} moebius(d) * C(n/d,2).
%C Zero followed by the Moebius transform of A000217. - _R. J. Mathar_, Jan 19 2009
%C Apparently, a(n-1) is the number of periodic complex Horadam orbits with period n, for n>2. - _Nathaniel Johnston_, Oct 04 2013
%C Also apparently, the first differences of A100448 (checked up to n=2000).
%H Alois P. Heinz, <a href="/A102309/b102309.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 Ovidiu Bagdasar, <a href="/A102309/a102309.pdf">On certain computational and geometric properties of complex Horadam orbits</a>, poster, ANTS 2014.
%H O. D. Bagdasar and P. 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 O. D. Bagdasar and P. 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 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 Ovidiu Bagdasar and I.-L. 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 G.f.: Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3. - _Seiichi Manyama_, May 24 2021
%p with(numtheory):
%p a:= n-> add(mobius(d)*binomial(n/d, 2), d=divisors(n)):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Feb 18 2013
%t a[n_] := Sum[MoebiusMu[d] Binomial[n/d, 2], {d, Divisors[n]}];
%t a /@ Range[0, 60] (* _Jean-François Alcover_, Feb 04 2020 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(d) * binomial(n/d,2) ); /* _Joerg Arndt_, Feb 18 2013 */
%o (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
%Y Second column of triangle A020921.
%Y Cf. A008683, A326419.
%K nonn,changed
%O 0,4
%A _Ralf Stephan_, Jan 03 2005