A080917 Number of integer solutions to the equation 2*x^2 + y^2 + 8*z^2 = n.

1, 2, 2, 4, 2, 0, 4, 0, 4, 10, 4, 12, 8, 0, 8, 0, 6, 16, 6, 12, 8, 0, 4, 0, 8, 10, 12, 16, 0, 0, 8, 0, 12, 16, 8, 24, 10, 0, 12, 0, 8, 32, 8, 12, 24, 0, 8, 0, 8, 18, 14, 24, 8, 0, 16, 0, 16, 16, 4, 36, 0, 0, 16, 0, 6, 32, 16, 12, 16, 0, 8, 0, 12, 16, 20, 28, 24, 0, 8, 0, 24, 34, 8, 36, 16, 0

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

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

Formula

Euler transform of period-32 sequence [2, -1, 2, -4, 2, -1, 2, 0, 2, -1, 2, -4, 2, -1, 2, -5, 2, -1, 2, -4, 2, -1, 2, 0, 2, -1, 2, -4, 2, -1, 2, -3, ...].

G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^8).

a(2*n - 1) = A072068(n). a(2*n) = A033717(n).

Example

G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 2*q^4 + 4*q^6 + 4*q^8 + 10*q^9 + 4*q^10 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^8], {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)

Prog

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

Crossrefs

Cf. A000122 (theta_3(q)), A033717, A072068, A080918.

Sequence in context: A033738 A033734 A084300 * A033726 A378751 A359382

Adjacent sequences: A080914 A080915 A080916 * A080918 A080919 A080920

Keyword

nonn

Author

Michael Somos, Feb 23 2003

Status

approved