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A318984
Theta series of quadratic form x^2 + x*y + 17*y^2.
5
1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 4, 0, 4, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 4, 0, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0
OFFSET
0,2
COMMENTS
Number of integer solutions (x, y) to x^2 + x*y + 17*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-67)].
LINKS
FORMULA
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-67, k) * x^k / (1 - x^k).
a(n) = 2 * A318982(n) unless n = 0.
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(67^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-67, p) = -1, b(p^e) = e + 1 if Kronecker(-67, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(67) = 0.767613... . - Amiram Eldar, Dec 16 2023
EXAMPLE
G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 4*x^17 + 4*x^19 + 4*x^23 + 2*x^25 + 4*x^29 + 2*x^36 + 4*x^37 + 4*x^47 + 2*x^49 + 4*x^59 + 2*x^64 + 2*x^67 + 4*x^68 + 4*x^71 + 4*x^73 + 4*x^76 + ...
MATHEMATICA
Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-67, #] &]];
2 Table[a[n], {n, 1, 110}]] (* Vincenzo Librandi, Sep 10 2018 *)
PROG
(PARI) a(n) = if(n, 2*sumdiv(n, d, kronecker(-67, d)), 1)
CROSSREFS
Cf. A318982.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43), this sequence (d=-67), A318985 (d=-163).
Sequence in context: A033735 A033731 A033729 * A227395 A255258 A329266
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Sep 06 2018
STATUS
approved