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A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).

%I A344817 #25 Jun 27 2024 01:38:43
%S A344817 1,0,5,-4,13,-16,49,-88,173,-324,701,-1384,2713,-5416,10989,-21916,
%T A344817 43621,-87224,174921,-349872,698773,-1397356,2796949,-5593872,
%U A344817 11183361,-22366976,44742149,-89483716,178951741,-357903312,715838513,-1431678040,2863290285
%N A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).
%H A344817 Seiichi Manyama, <a href="/A344817/b344817.txt">Table of n, a(n) for n = 1..1000</a>
%F A344817 a(n) = Sum_{k=1..n} Sum_{d|k} (-2)^(d-1).
%F A344817 G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 2*x^k).
%F A344817 G.f.: (1/(1 - x)) * Sum_{k>=1} (-2)^(k-1) * x^k/(1 - x^k).
%F A344817 a(n) ~ -(-1)^n * 2^n / 3. - _Vaclav Kotesovec_, Jun 05 2021
%t A344817 a[n_] := Sum[(-2)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* _Amiram Eldar_, May 29 2021 *)
%o A344817 (PARI) a(n) = sum(k=1, n, n\k*(-2)^(k-1));
%o A344817 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-2)^(d-1)));
%o A344817 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+2*x^k))/(1-x))
%o A344817 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-2)^(k-1)*x^k/(1-x^k))/(1-x))
%o A344817 (Magma)
%o A344817 A344817:= func< n | (&+[Floor(n/k)*(-2)^(k-1): k in [1..n]]) >;
%o A344817 [A344817(n): n in [1..40]]; // _G. C. Greubel_, Jun 26 2024
%o A344817 (SageMath)
%o A344817 def A344817(n): return sum((n//k)*(-2)^(k-1) for k in range(1,n+1))
%o A344817 [A344817(n) for n in range(1,41)] # _G. C. Greubel_, Jun 26 2024
%Y A344817 Column k=2 of A344824.
%Y A344817 Cf. A081295.
%Y A344817 Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), this sequence (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).
%K A344817 sign
%O A344817 1,3
%A A344817 _Seiichi Manyama_, May 29 2021