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A035147 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -43.

%I A035147 #25 Nov 18 2023 06:35:19
%S A035147 1,0,0,1,0,0,0,0,1,0,2,0,2,0,0,1,2,0,0,0,0,0,2,0,1,0,0,0,0,0,2,0,0,0,
%T A035147 0,1,0,0,0,0,2,0,1,2,0,0,2,0,1,0,0,2,2,0,0,0,0,0,2,0,0,0,0,1,0,0,2,2,
%U A035147 0,0,0,0,0,0,0,0,0,0,2,0,1
%N A035147 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -43.
%C A035147 From _Jianing Song_, Sep 07 2018: (Start)
%C A035147 Half of the number of integer solutions to x^2 + x*y + 11*y^2 = n.
%C A035147 Inverse Moebius transform of A011591. (End)
%H A035147 G. C. Greubel, <a href="/A035147/b035147.txt">Table of n, a(n) for n = 1..10000</a>
%F A035147 From _Jianing Song_, Sep 07 2018: (Start)
%F A035147 a(n) is multiplicative with a(43^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-43, p) = -1, a(p^e) = e + 1 if Kronecker(-43, p) = 1.
%F A035147 G.f.: Sum_{k>0} Kronecker(-43, k) * x^k / (1 - x^k).
%F A035147 A138811(n) = 2 * a(n) unless n = 0. (End)
%F A035147 From _Amiram Eldar_, Nov 18 2023: (Start)
%F A035147 a(n) = Sum_{d|n} Kronecker(-43, d).
%F A035147 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(43) = 0.479088... . (End)
%t A035147 a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-43, #] &]];
%t A035147 Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 25 2018 *)
%o A035147 (PARI) my(m=-43); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
%o A035147 (PARI) a(n) = sumdiv(n, d, kronecker(-43, d)); \\ _Amiram Eldar_, Nov 18 2023
%Y A035147 Cf. A138811.
%Y A035147 Moebius transform gives A011591.
%K A035147 nonn,easy,mult
%O A035147 1,11
%A A035147 _N. J. A. Sloane_