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A344818 a(n) = Sum_{k=1..n} floor(n/k) * (-3)^(k-1).

%I A344818 #26 Jun 27 2024 01:38:38
%S A344818 1,-1,9,-20,62,-174,556,-1660,4911,-14693,44357,-133053,398389,
%T A344818 -1195207,3587853,-10763270,32283452,-96850386,290570104,-871710994,
%U A344818 2615074146,-7845220010,23535839600,-70607518824,211822017739,-635466060265,1906399774635,-5719199303975
%N A344818 a(n) = Sum_{k=1..n} floor(n/k) * (-3)^(k-1).
%H A344818 Seiichi Manyama, <a href="/A344818/b344818.txt">Table of n, a(n) for n = 1..1000</a>
%F A344818 a(n) = Sum_{k=1..n} Sum_{d|k} (-3)^(d-1).
%F A344818 G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 3*x^k).
%F A344818 G.f.: (1/(1 - x)) * Sum_{k>=1} (-3)^(k-1) * x^k/(1 - x^k).
%F A344818 a(n) ~ -(-1)^n * 3^n / 4. - _Vaclav Kotesovec_, Jun 05 2021
%t A344818 a[n_] := Sum[(-3)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* _Amiram Eldar_, May 29 2021 *)
%o A344818 (PARI) a(n) = sum(k=1, n, n\k*(-3)^(k-1));
%o A344818 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-3)^(d-1)));
%o A344818 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+3*x^k))/(1-x))
%o A344818 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-3)^(k-1)*x^k/(1-x^k))/(1-x))
%o A344818 (Magma)
%o A344818 A344818:= func< n | (&+[Floor(n/k)*(-3)^(k-1): k in [1..n]]) >;
%o A344818 [A344818(n): n in [1..40]]; // _G. C. Greubel_, Jun 26 2024
%o A344818 (SageMath)
%o A344818 def A344818(n): return sum((n//k)*(-3)^(k-1) for k in range(1,n+1))
%o A344818 [A344818(n) for n in range(1,41)] # _G. C. Greubel_, Jun 26 2024
%Y A344818 Column k=3 of A344824.
%Y A344818 Cf. A101561.
%Y A344818 Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), this sequence (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).
%K A344818 sign
%O A344818 1,3
%A A344818 _Seiichi Manyama_, May 29 2021