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A080917 Number of integer solutions to the equation 2*x^2 + y^2 + 8*z^2 = n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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 A359382 A126768
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 23 2003
STATUS
approved

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)