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 A028641 #38 Feb 16 2025 08:32:35
%S A028641 1,2,0,0,2,4,0,4,0,2,0,4,0,0,0,0,2,4,0,2,4,0,0,4,0,6,0,0,4,0,0,0,0,0,
%T A028641 0,8,2,0,0,0,0,0,0,4,4,4,0,4,0,6,0,0,0,0,0,8,0,0,0,0,0,4,0,4,2,0,0,0,
%U A028641 4,0,0,0,0,4,0,0,2,8,0,0,4,2,0,4,0,8,0,0,0,0,0,0,4,0,0,4,0,0,0,4,6,4,0,0,0
%N A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.
%C A028641 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A028641 The number of integer solutions (x, y) to x^2 + x*y + 5*y^2 = n, discriminant -19. - _Ray Chandler_, Jul 12 2014
%D A028641 Robert Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 409, Eq. (19).
%H A028641 G. C. Greubel, <a href="/A028641/b028641.txt">Table of n, a(n) for n = 0..10000</a>
%H A028641 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references).
%H A028641 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A028641 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A028641 Theta series of quadratic form with Gram matrix [ 2, 1; 1, 10 ].
%F A028641 Expansion of phi(q) * phi(q^19) + 4 * q^5 * psi(q^2)* psi(q^38) in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Feb 27 2007
%F A028641 Moebius transform is period 19 sequence [2, -2, -2, 2, 2, 2, 2, -2, 2, -2, 2, -2, -2, -2, -2, 2, 2, -2, 0, ...]. - _Michael Somos_, Feb 27 2007
%F A028641 a(n) = 2*b(n) where b(n) is multiplicative with a(0) = 1, b(19^e) = 1, b(p^e) = e + 1 if Kronecker(-19, p) = 1, b(p^e) = (1 + (-1)^e)/2 if Kronecker(-19, p) = -1. - _Michael Somos_, Feb 27 2007
%F A028641 a(n) = 2 * A035171(n) unless n = 0. - _Jianing Song_, Sep 06 2018
%F A028641 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(19) = 1.441461... . - _Amiram Eldar_, Dec 16 2023
%e A028641 G.f. = 1 + 2*x + 2*x^4 + 4*x^5 + 4*x^7 + 2*x^9 + 4*x^11 + 2*x^16 + 4*x^17 + 2*x^19 + ...
%t A028641 a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -19, #] &] 2]; (* _Michael Somos_, Jun 14 2012 *)
%o A028641 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-19, d)) * 2)}; /* _Michael Somos_, Feb 27 2007 */
%o A028641 (PARI) {a(n) = if( n<1, n==0, qfrep([2, 1;1, 10], n, 1)[n] * 2)}; /* _Michael Somos_, Feb 27 2007 */
%Y A028641 Cf. A035171.
%Y A028641 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), this sequence (d=-19), A138811 (d=-43).
%K A028641 nonn,easy
%O A028641 0,2
%A A028641 _N. J. A. Sloane_