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A344819 a(n) = Sum_{k=1..n} floor(n/k) * (-4)^(k-1).

%I A344819 #25 Jun 25 2024 19:25:49
%S A344819 1,-2,15,-52,205,-806,3291,-13160,52393,-209498,839079,-3356300,
%T A344819 13420917,-53683854,214751875,-859006400,3435960897,-13743843762,
%U A344819 54975632975,-219902535924,879609095965,-3518436366566,14073749677851,-56294998711576,225179977999337,-900719912066074
%N A344819 a(n) = Sum_{k=1..n} floor(n/k) * (-4)^(k-1).
%H A344819 Seiichi Manyama, <a href="/A344819/b344819.txt">Table of n, a(n) for n = 1..1000</a>
%F A344819 a(n) = Sum_{k=1..n} Sum_{d|k} (-4)^(d-1).
%F A344819 G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 4*x^k).
%F A344819 G.f.: (1/(1 - x)) * Sum_{k>=1} (-4)^(k-1) * x^k/(1 - x^k).
%F A344819 a(n) ~ -(-1)^n * 4^n / 5. - _Vaclav Kotesovec_, Jun 05 2021
%t A344819 a[n_] := Sum[(-4)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* _Amiram Eldar_, May 29 2021 *)
%o A344819 (PARI) a(n) = sum(k=1, n, n\k*(-4)^(k-1));
%o A344819 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-4)^(d-1)));
%o A344819 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+4*x^k))/(1-x))
%o A344819 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-4)^(k-1)*x^k/(1-x^k))/(1-x))
%o A344819 (Magma)
%o A344819 A344819:= func< n | (&+[(-4)^(k-1)*Floor(n/k): k in [1..n]]) >;
%o A344819 [A344819(n): n in [1..40]]; // _G. C. Greubel_, Jun 25 2024
%o A344819 (SageMath)
%o A344819 def A344819(n): return sum((-4)^(k-1)*int(n//k) for k in range(1,n+1))
%o A344819 [A344819(n) for n in range(1,41)] # _G. C. Greubel_, Jun 25 2024
%Y A344819 Column k=4 of A344824.
%Y A344819 Cf. A059851, A101562, A268235, A332533, A344814, A344815, A344816, A344817, A344818, A344820.
%K A344819 sign
%O A344819 1,2
%A A344819 _Seiichi Manyama_, May 29 2021