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G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 + x)^2.
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%I #17 Sep 08 2022 08:46:24

%S 1,-1,3,-5,5,-3,7,-13,9,-5,11,-15,13,-7,15,-29,17,-9,19,-25,21,-11,23,

%T -39,25,-13,27,-35,29,-15,31,-61,33,-17,35,-45,37,-19,39,-65,41,-21,

%U 43,-55,45,-23,47,-87,49,-25,51,-65,53,-27,55,-91,57,-29,59,-75,61,-31,63,-125,65

%N G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 + x)^2.

%F G.f.: Sum_{k>=0} x^(2^k) / (1 + x^(2^k))^2.

%F G.f.: Sum_{k>=1} (-1)^(k + 1) * phi(2*k) * x^k / (1 - x^k), where phi = A000010.

%F a(2*n) = a(n) - 2*n, a(2*n+1) = 2*n + 1.

%F From _Werner Schulte_, Oct 05 2020: (Start)

%F Multiplicative with a(2^e) = 3 - 2^(e+1) and a(p^e) = p^e for e >= 0 and prime p > 2.

%F Dirichlet g. f.: Sum_{n>0} a(n)/n^s = zeta(s-1) * (1-3/(2^s-1)). (End)

%t nmax = 65; CoefficientList[Series[Sum[x^(2^k)/(1 + x^(2^k))^2, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest

%t nmax = 65; CoefficientList[Series[Sum[(-1)^(k + 1) EulerPhi[2 k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t a[n_] := If[EvenQ[n], a[n/2] - n, n]; Table[a[n], {n, 1, 65}]

%o (Magma) a:=[1]; for k in [1..65] do if IsOdd(k) then a[k]:=k; else a[k]:=a[k div 2]-k; end if; end for; a; // _Marius A. Burtea_, Oct 07 2019

%o (PARI) a(n) = if (n==1, 1, if (n % 2, n, a(n/2) - n)); \\ _Michel Marcus_, Oct 07 2019

%o (PARI) a(n) = 3*(n>>valuation(n,2)) - n<<1; \\ _Kevin Ryde_, Oct 06 2020

%Y Cf. A000010, A001511, A062570, A088705, A129527.

%K sign,mult

%O 1,3

%A _Ilya Gutkovskiy_, Oct 05 2019