A350163 - OEIS

%I #18 Dec 18 2021 11:17:44

%S 1,8,26,63,125,209,335,504,703,981,1311,1671,2141,2681,3269,3990,4808,

%T 5643,6669,7847,8963,10343,11861,13349,15212,17170,19078,21310,23748,

%U 26172,28962,31939,34759,38133,41769,45190,49188,53400,57396,62246,67168,71704,77122

%N a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^3.

%F a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (d^3 - (d - 1)^3).

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

%t a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^3, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Dec 18 2021 *)

%o (PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^3);

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(d^3-(d-1)^3)));

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1+x^(2*k)))/(1-x))

%Y Column 3 of A350161.

%Y Cf. A101455, A344721, A350144.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Dec 18 2021