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a(n) = Sum_{d|n} mu(n/d) * floor(d^2/4).
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%I #9 Aug 03 2021 15:25:46

%S 0,1,2,3,6,6,12,12,18,18,30,24,42,36,48,48,72,54,90,72,96,90,132,96,

%T 150,126,162,144,210,144,240,192,240,216,288,216,342,270,336,288,420,

%U 288,462,360,432,396,552,384,588,450,576,504,702,486,720,576,720,630,870,576,930

%N a(n) = Sum_{d|n} mu(n/d) * floor(d^2/4).

%C Moebius transform of quarter-squares (A002620).

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

%F a(n) = J_2(n) / 4 for n >= 3, where J_() is the Jordan function.

%t Table[Sum[MoebiusMu[n/d] Floor[d^2/4], {d, Divisors[n]}], {n, 1, 61}]

%t nmax = 61; CoefficientList[Series[Sum[MoebiusMu[k] x^(2 k)/((1 + x^k) (1 - x^k)^3), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(d^2\4)); \\ _Michel Marcus_, Aug 03 2021

%Y Cf. A002620, A007434, A023022, A346759.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Aug 02 2021