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A033716
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Number of integer solutions to the equation x^2 + 3y^2 = n.
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18
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1, 2, 0, 2, 6, 0, 0, 4, 0, 2, 0, 0, 6, 4, 0, 0, 6, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 2, 12, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 6, 6, 0, 0, 12, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 12, 0, 0, 4, 0, 2, 0, 0, 12, 0, 0, 0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 4, 0, 0, 6, 0
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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)
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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A033716 := proc(n)
local a, j ;
a := 0 ;
for j from 0 to n/3 do
a := a+A000122(n-3*j)*A000122(j) ;
end do:
a;
end proc:
seq(A033716(n), n=0..80) ; # R. J. Mathar, Feb 22 2021
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A096936, A119395, A129576.
Sequence in context: A339327 A258144 A113772 * A115978 A033751 A033745
Adjacent sequences: A033713 A033714 A033715 * A033717 A033718 A033719
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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