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

1, 2, 4, 7, 12, 26, 44, 92, 175, 354, 684, 1396, 2732, 5506, 10938, 21937, 43692, 87578, 174764, 349884, 699098, 1398786, 2796204, 5593886, 11184823, 22372354, 44739418, 89483996, 178956972, 357925242, 715827884, 1431677702, 2863312218, 5726666754, 11453246178, 22906581193

Offset

1,2

Comments

Inverse Moebius transform of Jacobsthal numbers (A001045).

Links

Table of n, a(n) for n=1..36.

Formula

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

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

Maple

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

Mathematica

nmax = 36; CoefficientList[Series[Sum[x^k/(1 - x^k - 2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[(1/3) Sum[(2^d - (-1)^d), {d, Divisors[n]}], {n, 1, 36}]

Prog

(PARI) a(n)={sumdiv(n, d, 2^d - (-1)^d)/3} \\ Andrew Howroyd, Aug 14 2019
(Python)
n = 1
while n <= 36:
    s, d = 0, 1
    while d <= n:
        if n%d == 0:
            s = s+2**d-(-1)**d
        d = d+1
    print(n, s//3)
n = n+1 # A.H.M. Smeets, Aug 14 2019

Crossrefs

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

Sequence in context: A332836 A328129 A360890 * A027945 A079800 A217595

Adjacent sequences: A309726 A309727 A309728 * A309730 A309731 A309732

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 14 2019

Status

approved