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A309729 Expansion of Sum_{k>=1} x^k/(1 - x^k - 2*x^(2*k)).

%I A309729 #11 Aug 14 2019 22:19:05
%S A309729 1,2,4,7,12,26,44,92,175,354,684,1396,2732,5506,10938,21937,43692,
%T A309729 87578,174764,349884,699098,1398786,2796204,5593886,11184823,22372354,
%U A309729 44739418,89483996,178956972,357925242,715827884,1431677702,2863312218,5726666754,11453246178,22906581193
%N A309729 Expansion of Sum_{k>=1} x^k/(1 - x^k - 2*x^(2*k)).
%C A309729 Inverse Moebius transform of Jacobsthal numbers (A001045).
%F A309729 G.f.: Sum_{k>=1} A001045(k) * x^k/(1 - x^k).
%F A309729 a(n) = (1/3) * Sum_{d|n} (2^d - (-1)^d).
%p A309729 seq(add(2^d-(-1)^d, d=numtheory:-divisors(n))/3, n=1..50); # _Robert Israel_, Aug 14 2019
%t A309729 nmax = 36; CoefficientList[Series[Sum[x^k/(1 - x^k - 2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A309729 Table[(1/3) Sum[(2^d - (-1)^d), {d, Divisors[n]}], {n, 1, 36}]
%o A309729 (PARI) a(n)={sumdiv(n, d, 2^d - (-1)^d)/3} \\ _Andrew Howroyd_, Aug 14 2019
%o A309729 (Python)
%o A309729 n = 1
%o A309729 while n <= 36:
%o A309729     s, d = 0, 1
%o A309729     while d <= n:
%o A309729         if n%d == 0:
%o A309729             s = s+2**d-(-1)**d
%o A309729         d = d+1
%o A309729     print(n,s//3)
%o A309729 n = n+1 # _A.H.M. Smeets_, Aug 14 2019
%Y A309729 Cf. A001045, A007435, A055895, A100107, A104723, A256281.
%K A309729 nonn
%O A309729 1,2
%A A309729 _Ilya Gutkovskiy_, Aug 14 2019