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A034896
Number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.
10
1, 4, 4, 4, 20, 24, 4, 32, 52, 4, 24, 48, 20, 56, 32, 24, 116, 72, 4, 80, 120, 32, 48, 96, 52, 124, 56, 4, 160, 120, 24, 128, 244, 48, 72, 192, 20, 152, 80, 56, 312, 168, 32, 176, 240, 24, 96, 192, 116, 228, 124, 72, 280, 216, 4, 288, 416, 80, 120, 240, 120, 248, 128, 32, 500
OFFSET
0,2
COMMENTS
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 16 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223, Entry 3(iv).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
J. Liouville, Sur la forme x^2 + y^2 + 3(z^2 + t^2), Journal de mathématiques pures et appliquées 2e série, tome 5 (1860), p. 147-152.
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
FORMULA
Expansion of theta_3(q)^2*theta_3(q^3)^2.
G.f.: s(2)^10*s(6)^10/(s(1)*s(3)*s(4)*s(12))^4, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
Fine gives an explicit formula for a(n) in terms of the divisors of n.
From Michael Somos, Nov 10 2018: (Start)
Expansion of (a(q) + 2*a(q^4))^2 / 9 = (a(q)^2 - 2*a(q^2)^2 + 4*a(q^4)^2) / 3 in powers of q where a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + 4 Sum_{k>0} k x^k / (1 - (-x)^k) Kronecker(9, k).
a(n) = 1 + 4 * A113262(n) = (-1)^n * A134946(n). Convolution square of A033716.
a(n) = 4 * (s(n) - 2*s(n/2) - 3*s(n/3) + 4*s(n/4) + 6*s(n/6) - 12*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0. (End)
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/12) * Gamma(2/3)^2*3^(1/2) * (1+3^(1/2))^2 / Gamma(11/12)^2 / Gamma(3/4)^2 = A388397. - Simon Plouffe, Sep 15 2025
EXAMPLE
G.f. = 1 + 4*x + 4*x^2 + 4*x^3 + 20*x^4 + 24*x^5 + 4*x^6 + 32*x^7 + ... - Michael Somos, Nov 10 2018
MATHEMATICA
A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* Michael Somos, Nov 10 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^10 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^4, n))}; /* Michael Somos, Nov 10 2018 */
CROSSREFS
KEYWORD
nonn
STATUS
approved