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 A004531 #40 Feb 16 2025 08:32:28
%S A004531 1,2,0,0,4,4,0,0,4,2,0,0,0,4,0,0,4,4,0,0,8,0,0,0,0,6,0,0,0,4,0,0,4,0,
%T A004531 0,0,4,4,0,0,8,4,0,0,0,4,0,0,0,2,0,0,8,4,0,0,0,0,0,0,0,4,0,0,4,8,0,0,
%U A004531 8,0,0,0,4,4,0,0,0,0,0,0,8,2,0,0,0,8,0,0,0,4,0,0,0,0,0,0,0,4,0,0,12,4,0,0,8
%N A004531 Number of integer solutions to x^2 + 4 * y^2 = n.
%C A004531 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A004531 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
%D A004531 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 32.
%H A004531 Seiichi Manyama, <a href="/A004531/b004531.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H A004531 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A004531 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A004531 Expansion of (eta(q^2) * eta(q^8))^5 / (eta(q)^2 * eta(q^4)^4 * eta(q^16)^2) in powers of q.
%F A004531 Expansion of phi(x) * phi(x^4) = phi(x^4)^2 + 2 * x * psi(x^4)^2 in powers of x where phi(x), psi(x) are Ramanujan theta functions.
%F A004531 Expansion of (theta2^2(q^2) + theta3^2(q^2) + theta4^2(q^2)) / 2 in powers of q.
%F A004531 Euler transform of period 16 sequence [ 2, -3, 2, 1, 2, -3, 2, -4, 2, -3, 2, 1, 2, -3, 2, -2, ...]. - _Michael Somos_, Jun 20 2014
%F A004531 G.f.: Sum_{i,j} x^(i^2 + 4 * j^2).
%F A004531 a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A004018(n). a(4*n + 1) = A004020(n).
%F A004531 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - _Amiram Eldar_, Oct 15 2022
%e A004531 G.f. = 1 + 2*x + 4*x^4 + 4*x^5 + 4*x^8 + 2*x^9 + 4*x^13 + 4*x^16 + 4*x^17 + 8*x^20 + ...
%t A004531 CoefficientList[EllipticTheta[3, 0, x]*EllipticTheta[3, 0, x^4] + O[x]^105, x] (* _Jean-François Alcover_, Nov 05 2015 *)
%o A004531 (PARI) {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n])}; /* _Michael Somos_, Jul 04 2005 */
%o A004531 (PARI) {a(n) = local(A, e1, e2, e4); if( n<0, 0, A = x * O(x^n); e1 = eta(x^2 + A); e2 = eta(x^4 + A); e4 = eta(x^8 + A); polcoeff( (e2^12 + e1^8 * e4^4 + 4 * x * e1^4 * e4^8) / (2 * e1^4 * e2^2 * e4^4), n))};
%o A004531 (Sage)
%o A004531 Q = DiagonalQuadraticForm(ZZ, [4, 1])
%o A004531 Q.representation_number_list(105) # _Peter Luschny_, Jun 20 2014
%Y A004531 Cf. A004018, A004020, A019669.
%K A004531 nonn
%O A004531 0,2
%A A004531 _N. J. A. Sloane_