A072070 Number of integer solutions to the equation 4*x^2 + y^2 + 8*z^2 = n.

1, 2, 0, 0, 4, 4, 0, 0, 6, 6, 0, 0, 8, 12, 0, 0, 12, 8, 0, 0, 8, 8, 0, 0, 8, 14, 0, 0, 16, 4, 0, 0, 6, 16, 0, 0, 12, 20, 0, 0, 24, 8, 0, 0, 8, 20, 0, 0, 24, 18, 0, 0, 24, 12, 0, 0, 0, 16, 0, 0, 16, 20, 0, 0, 12, 8, 0, 0, 16, 16, 0, 0, 30, 32, 0, 0, 24, 16, 0, 0, 24, 18, 0, 0, 16, 24, 0, 0, 24, 16

Offset: 0

Author

T. D. Noe, Jun 13 2002

Keywords

• nonn

Comments

Related to primitive congruent numbers A006991.

Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and a(n) = 2 A072071(n).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			a(4) = 4 because (1, 0, 0), (-1, 0, 0), (0, 2, 0) and (0, -2, 0) are solutions.
G.f. = 1 + 2*q + 4*q^4 + 4*q^5 + 6*q^8 + 6*q^9 + 8*q^12 + 12*q^13 + 12*q^16 + 8*q^17 + ...
		

References

• J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture
• Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem
• Karl Rubin, Elliptic curves and right triangles
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A006991, A003273, A072068, A072069, A072071, A080917.

Programs

Mathematica

maxN=128; soln3=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/8]]; yMax=Ceiling[Sqrt[maxN/2]]; zMax=Ceiling[Sqrt[maxN/16]]; Do[n=4x^2+y^2+8z^2; If[n>0&&n<=maxN/2, s=8; If[x==0, s=s/2]; If[y==0, s=s/2]; If[z==0, s=s/2]; soln3[[n]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4] EllipticTheta[ 3, 0, q^8], {q, 0, n}]; (* Michael Somos, Jul 23 2018 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-4 * eta(x^8 + A)^3 * eta(x^16 + A)^3 * eta(x^32 + A)^-2, n))}; /* Michael Somos, Feb 11 2003 */

Formula

Expansion of phi(q) * phi(q^4) * phi(q^8) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 09 2012

Euler transform of period 32 sequence [2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -5, 2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -3, ...]. - Michael Somos, Feb 11 2003

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A014455(n). - Michael Somos, Jun 08 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A080917. - Michael Somos, Jul 23 2018