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 A138811 #32 Feb 16 2025 08:33:08
%S A138811 1,2,0,0,2,0,0,0,0,2,0,4,0,4,0,0,2,4,0,0,0,0,0,4,0,2,0,0,0,0,0,4,0,0,
%T A138811 0,0,2,0,0,0,0,4,0,2,4,0,0,4,0,2,0,0,4,4,0,0,0,0,0,4,0,0,0,0,2,0,0,4,
%U A138811 4,0,0,0,0,0,0,0,0,0,0,4,0,2,0,4,0,0,0
%N A138811 Theta series of quadratic form x^2 + x*y + 11*y^2.
%C A138811 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138811 G. C. Greubel, <a href="/A138811/b138811.txt">Table of n, a(n) for n = 0..10000</a>
%H A138811 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A138811 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A138811 Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
%F A138811 Expansion of phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A138811 Moebius transform is period 43 sequence [2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
%F A138811 a(n) = 2*b(n) where b() is multiplicative with b(43^e) = 1, b(p^e) = e + 1 if Kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 otherwise.
%F A138811 G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A138811 a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
%F A138811 G.f.: Sum_{i,j in Z} x^(i*i + i*j + 11*j*j).
%F A138811 a(n) = 2 * A035147(n) unless n = 0.
%F A138811 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(43) = 0.958176... . - _Amiram Eldar_, Nov 21 2023
%e A138811 G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...
%t A138811 a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -43, #] &]]; (* _Michael Somos_, Sep 07 2015 *)
%t A138811 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^43] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^43], {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t A138811 Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-43, #]&]];
%t A138811 2 Table[a[n], {n, 1, 100}]] (* _Vincenzo Librandi_, Sep 07 2018 *)
%o A138811 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)};
%o A138811 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))};
%o A138811 (PARI) a(n)=if(n, sumdivmult(n,d,kronecker(-43,d))*2, 0) \\ _Charles R Greathouse IV_, Nov 23 2021
%o A138811 (Magma) A := Basis( ModularForms( Gamma1(43), 1), 87); A[1] + 2*A[2] + 2*A[5] + 2*A[10] + 4*A[12] + 4*A[14] + 2*A[17] + 4*A[18]; /* _Michael Somos_, Sep 07 2015 */
%Y A138811 Cf. A035147.
%Y A138811 Cf. A000122, A000700, A010054, A121373.
%Y A138811 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), this sequence (d=-43).
%K A138811 nonn,easy
%O A138811 0,2
%A A138811 _Michael Somos_, Mar 31 2008, Apr 05 2008