This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064027 #43 Sep 21 2023 01:45:10
%S A064027 1,3,10,19,26,30,50,83,91,78,122,190,170,150,260,339,290,273,362,494,
%T A064027 500,366,530,830,651,510,820,950,842,780,962,1363,1220,870,1300,1729,
%U A064027 1370,1086,1700,2158,1682,1500,1850,2318,2366,1590,2210,3390,2451,1953
%N A064027 a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.
%D A064027 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
%H A064027 Paul D. Hanna, <a href="/A064027/b064027.txt">Table of n, a(n) for n = 1..1000</a>
%H A064027 Heekyoung Hahn, <a href="https://arxiv.org/abs/1507.04426">Convolution sums of some functions on divisors</a>, arXiv:1507.04426 [math.NT], 2015.
%F A064027 Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
%F A064027 a(n) = (-1)^n*(A001157(n) - 2*A050999(n)).
%F A064027 Logarithmic derivative of A224364. - _Paul D. Hanna_, Apr 04 2013
%F A064027 Bisection: a(2*k-1) = A001157(2*k-1), a(2*k) = 4*A001157(k) - A050999(2*k), k >= 1. In the Hardy reference a(n) = sigma^*_2(n). - _Wolfdieter Lang_, Jan 07 2017
%F A064027 G.f.: Sum_{k>=1} k^2*x^k/(1 - (-x)^k). - _Ilya Gutkovskiy_, Nov 09 2018
%F A064027 Sum_{k=1..n} a(k) ~ 7 * zeta(3) * n^3 / 24. - _Vaclav Kotesovec_, Nov 10 2018
%F A064027 Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 1/2^(s-1) + 1/2^(2*s-3)). - _Amiram Eldar_, Sep 21 2023
%e A064027 L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + ...
%e A064027 where exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 + 106*x^8 + 181*x^9 + ... + A224364(n)*x^n + ... - _Paul D. Hanna_, Apr 04 2013
%t A064027 a[n_] := (-1)^n DivisorSum[n, (-1)^# * #^2 &]; Array[a, 50] (* _Jean-François Alcover_, Dec 23 2015 *)
%t A064027 a[n_] := If[OddQ[n], 1, (1 - 6/(4^(IntegerExponent[n, 2] + 1) - 1))] * DivisorSigma[2, n]; Array[a, 100] (* _Amiram Eldar_, Sep 21 2023 *)
%o A064027 (PARI) {a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ _Paul D. Hanna_, Apr 04 2013
%o A064027 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // _G. C. Greubel_, Nov 09 2018
%Y A064027 Cf. A001157, A002129, A008457, A050999, A224364.
%Y A064027 Cf. A321543 - A321565, A321807 - A321836 for related sequences.
%K A064027 mult,easy,nonn
%O A064027 1,2
%A A064027 _Vladeta Jovovic_, Sep 11 2001