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A344720 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^2.

%I A344720 #28 May 28 2021 17:17:46
%S A344720 1,3,9,12,22,30,44,48,71,83,105,115,141,157,201,206,240,266,304,318,
%T A344720 378,402,448,460,519,547,623,641,699,747,809,815,907,943,1035,1064,
%U A344720 1138,1178,1286,1302,1384,1448,1534,1560,1710,1758,1852,1866,1977,2039,2179,2209,2315,2395,2535
%N A344720 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^2.
%H A344720 Seiichi Manyama, <a href="/A344720/b344720.txt">Table of n, a(n) for n = 1..10000</a>
%F A344720 a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (2*d - 1).
%F A344720 G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 + x^k).
%F A344720 a(n) ~ Pi^2 * n^2 / 12. - _Vaclav Kotesovec_, May 28 2021
%t A344720 a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^2, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 27 2021 *)
%t A344720 Accumulate[Table[-2*DivisorSigma[0, 2*n] + 3*DivisorSigma[0, n] + 2*DivisorSigma[1, 2*n] - 4*DivisorSigma[1, n], {n, 1, 50}]] (* _Vaclav Kotesovec_, May 28 2021 *)
%o A344720 (PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^2);
%o A344720 (PARI) a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(2*d-1)));
%o A344720 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1+x^k))/(1-x))
%Y A344720 Column k=2 of A344726.
%Y A344720 Cf. A059851, A078471, A222548.
%K A344720 nonn
%O A344720 1,2
%A A344720 _Seiichi Manyama_, May 27 2021