A309729 - OEIS

%I #11 Aug 14 2019 22:19:05

%S 1,2,4,7,12,26,44,92,175,354,684,1396,2732,5506,10938,21937,43692,

%T 87578,174764,349884,699098,1398786,2796204,5593886,11184823,22372354,

%U 44739418,89483996,178956972,357925242,715827884,1431677702,2863312218,5726666754,11453246178,22906581193

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

%C Inverse Moebius transform of Jacobsthal numbers (A001045).

%F G.f.: Sum_{k>=1} A001045(k) * x^k/(1 - x^k).

%F a(n) = (1/3) * Sum_{d|n} (2^d - (-1)^d).

%p seq(add(2^d-(-1)^d, d=numtheory:-divisors(n))/3, n=1..50); # _Robert Israel_, Aug 14 2019

%t nmax = 36; CoefficientList[Series[Sum[x^k/(1 - x^k - 2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[(1/3) Sum[(2^d - (-1)^d), {d, Divisors[n]}], {n, 1, 36}]

%o (PARI) a(n)={sumdiv(n, d, 2^d - (-1)^d)/3} \\ _Andrew Howroyd_, Aug 14 2019

%o (Python)

%o n = 1

%o while n <= 36:

%o     s, d = 0, 1

%o     while d <= n:

%o         if n%d == 0:

%o             s = s+2**d-(-1)**d

%o         d = d+1

%o     print(n,s//3)

%o n = n+1 # _A.H.M. Smeets_, Aug 14 2019

%Y Cf. A001045, A007435, A055895, A100107, A104723, A256281.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Aug 14 2019