A329970 - OEIS

%I #11 Sep 20 2021 22:14:34

%S 0,0,-2,3,0,-3,-7,16,2,-15,-21,31,24,-15,-57,34,25,-17,-27,77,8,-99,

%T -111,155,117,-36,-140,40,25,-80,-96,259,112,-157,-249,202,183,-156,

%U -354,224,203,-40,-62,342,-21,-524,-548,562,488,-34,-358,194,167,-262

%N a(n) = (-1)^(n + 1) * n * ceiling(n/2) + Sum_{k=1..n} (-1)^k * k^2 * floor(n/k).

%H Amiram Eldar, <a href="/A329970/b329970.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: x * (1 - x + 2*x^2) / ((1 - x)^2 * (1 + x)^3) + (1/(1 - x)) * Sum_{k>=1} (-1)^k * k^2 * x^k / (1 - x^k).

%F a(n) = Sum_{k=1..n} (-1)^(k + 1) * (n mod k) * k.

%t Table[(-1)^(n + 1) n Ceiling[n/2] + Sum[(-1)^k k^2 Floor[n/k], {k, 1, n}], {n, 1, 54}]

%t nmax = 54; CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x)^2 (1 + x)^3) + 1/(1 - x) Sum[(-1)^k k^2 x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[Sum[(-1)^(k + 1) Mod[n, k] k, {k, 1, n}], {n, 1, 54}]

%o (PARI) a(n) = (-1)^(n + 1)*n*ceil(n/2) + sum(k=1, n, (-1)^k * k^2 * (n\k)); \\ _Michel Marcus_, Sep 20 2021

%Y Cf. A004125, A051126, A093005, A154585, A309176.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, Nov 26 2019