A113262 One quarter of the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

1, 1, 1, 5, 6, 1, 8, 13, 1, 6, 12, 5, 14, 8, 6, 29, 18, 1, 20, 30, 8, 12, 24, 13, 31, 14, 1, 40, 30, 6, 32, 61, 12, 18, 48, 5, 38, 20, 14, 78, 42, 8, 44, 60, 6, 24, 48, 29, 57, 31, 18, 70, 54, 1, 72, 104, 20, 30, 60, 30, 62, 32, 8, 125, 84, 12, 68, 90, 24, 48, 72, 13, 74, 38, 31

Offset: 1

Author

Michael Somos, Oct 21 2005

Keywords

• nonn
• mult

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A034896, A222171.

Programs

Mathematica

A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n]/4, {n, 1, 50}] (* G. C. Greubel, Dec 24 2017 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* Michael Somos, Nov 10 2018 *)

PARI

{a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(9, d) * (-1)^(n-d)))};

PARI

{a(n) = my(A, p, e); if(n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==3, 1, (p^(e+1) - 1) / (p - 1) - 2*(p==2))))};

Formula

a(n) is multiplicative with a(3^e) = 1, a(2^e) = 2^(e+1) - 3, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 3.

G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k) * Kronecker(9, k) = ((theta_3(x) * theta_3(x^3))^2 - 1) / 4.

A034896(n) = 4*a(n) if n > 0.

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Dec 01 2022

Dirichlet g.f.: zeta(s)*zeta(s-1)*(4^(1-s)-2^(1-s)+1)*(1-3^(1-s)). - Amiram Eldar, Jan 06 2023