cp's OEIS Frontend

A318985 Theta series of quadratic form x^2 + x*y + 41*y^2.

%I A318985 #20 Dec 16 2023 09:01:31
%S A318985 1,2,0,0,2,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,
%T A318985 0,0,2,0,0,0,0,4,0,4,0,0,0,4,0,2,0,0,0,4,0,0,0,0,0,0,0,4,0,0,2,0,0,0,
%U A318985 0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0
%N A318985 Theta series of quadratic form x^2 + x*y + 41*y^2.
%C A318985 Number of integer solutions (x, y) to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)].
%H A318985 Jianing Song, <a href="/A318985/b318985.txt">Table of n, a(n) for n = 0..10000</a>
%H A318985 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a>.
%F A318985 G.f.: 1 + 2 * Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k).
%F A318985 a(n) = 2 * A318983(n) unless n = 0.
%F A318985 a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(163^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, b(p^e) = e + 1 if Kronecker(-163, p) = 1.
%F A318985 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(163) = 0.49213705... . - _Amiram Eldar_, Dec 16 2023
%e A318985 G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + 4*x^41 + 4*x^43 + 4*x^47 + 2*x^49 + 4*x^53 + 4*x^61 + 2*x^64 + 4*x^71 + ...
%t A318985 Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-163, #] &]];
%t A318985 2 Table[a[n], {n, 1, 110}]] (* _Vincenzo Librandi_, Sep 10 2018 *)
%o A318985 (PARI) a(n) = if(n, 2*sumdiv(n, d, kronecker(-163, d)), 1)
%Y A318985 Cf. A318983.
%Y A318985 Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43), A318984 (d=-67), this sequence (d=-163).
%K A318985 nonn,easy
%O A318985 0,2
%A A318985 _Jianing Song_, Sep 06 2018