OFFSET
0,2
COMMENTS
The cubic modular equation for k is equivalent to theta_4(q) * theta_4(q^3) + theta_2(q)* theta_2(q^3) = theta_3(q) * theta_3(q^3). - Michael Somos, Feb 17 2003
The number of nonnegative solutions is given by A119395. - Max Alekseyev, May 16 2006
Fermat used infinite descent to prove "That there is no number, less by a unit than a multiple of 3, which is composed of a square and the triple of another square". [Yves Hellegouarch, "Invitation to the Mathematics of Fermat-Wiles", Academic Press, 2002, page 4]. - Michael Somos, Sep 03 2016
REFERENCES
J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 110.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..65537
G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
Michael Gilleland, Some Self-Similar Integer Sequences.
M. D. Hirschhorn, Three classical results on representations of a number, Séminaire Lotharingien de Combinatoire, B42f (1999), 8 pp.
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
H. Movasati and Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2017.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
FORMULA
Fine gives an explicit formula for a(n) in terms of the divisors of n.
Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+3*j^2).
G.f.: s(2)^5*s(6)^5/(s(1)^2*s(3)^2*s(4)^2*s(12)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
Euler transform of period 12 sequence [ 2, -3, 4, -1, 2, -6, 2, -1, 4, -3, 2, -2, ...]. - Michael Somos, Feb 17 2003
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u1, u3, u9) = (u1*u9) * (u1^2 - 3*u1*u3 + 3*u3^2) * (u3^2 - 3*u3*u9 + 3*u9^2) - u3^6. - Michael Somos, Sep 05 2005
G.f.: theta_3(q) * theta_3(q^3) = (Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(3k^2)). - Michael Somos, Sep 05 2005
Let n=3^d*p1^(2*b1)*...*pm^(2*bm)*q1^c1*...*qk^ck be a prime factorization of n where pi are primes of the form 3t+2 and qj are primes of the form 3t+1. Let B=(c1+1)*...*(ck+1). Then a(n)=0 if either of bi is a half-integer; a(n)=6B if n is a multiple of 4; and a(n)=2B otherwise. - Max Alekseyev, May 16 2006
a(n) = 2 * A096936(n).
a(3*n + 2) = 0. a(3*n) = a(n). a(3*n + 1) = 2 * A129576(n). - Michael Somos, Sep 03 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Oct 15 2022
EXAMPLE
G.f. = 1 + 2*q + 2*q^3 + 6*q^4 + 4*q^7 + 2*q^9 + 6*q^12 + 4*q^13 + 6*q^16 + ...
MAPLE
A033716 := proc(n)
local a, j ;
a := 0 ;
for j from 0 to n/3 do
end do:
a;
end proc:
seq(A033716(n), n=0..80) ; # R. J. Mathar, Feb 22 2021
MATHEMATICA
a[n_] := With[{r = Reduce[x^2 + 3*y^2 == n, {x, y}, Integers]}, Which[r === False, 0, Head[r] === And, 1, True, Length[r]]]; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Jan 10 2014 *)
QP = QPochhammer; s = (QP[q^2] * QP[q^6])^5 / (QP[q] * QP[q^3] * QP[q^4] * QP[q^12])^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)
a[ n_] := Length @ FindInstance[ x^2 + 3 y^2 == n, {x, y}, Integers, 10^9]; (* Michael Somos, Sep 03 2016 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* Michael Somos, Sep 03 2016 *)
PROG
(PARI) {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 3], n)[n] * 2)}; /* Michael Somos, Jun 05 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^5 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^2, n))}; /* Michael Somos, Jun 05 2005 */
(PARI) { a(n) = local(f, B); f=factorint(n); B=1; for(i=1, matsize(f)[1], if(f[i, 1]%3==1, B*=f[i, 2]+1); if(f[i, 1]%3==2, if(f[i, 2]%2, return(0)))); if(n%4, 2*B, 6*B) } \\ Max Alekseyev, May 16 2006
(PARI) first(n) = {my(res = vector(n + 1)); for(i = 0, sqrtint(n \ 3), for(j = 0, sqrtint(n - 3*i^2), res[3*i^2 + j^2 + 1] += (1<<(!!i + !!j)))); res} \\ David A. Corneth, Nov 20 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved