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

0,2

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).

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

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A329264 A045836 A182056 * A137830 A137828 A264655

Adjacent sequences: A072067 A072068 A072069 * A072071 A072072 A072073

Keyword

nonn

Author

T. D. Noe, Jun 13 2002

Status

approved