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
Keywords
Programs
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Maple
seq(add(2^d-(-1)^d, d=numtheory:-divisors(n))/3, n=1..50); # Robert Israel, Aug 14 2019
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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}] -
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
Formula
G.f.: Sum_{k>=1} A001045(k) * x^k/(1 - x^k).
a(n) = (1/3) * Sum_{d|n} (2^d - (-1)^d).
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