The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A318985 Theta series of quadratic form x^2 + x*y + 41*y^2. 5
 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of integer solutions (x, y) to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)]. LINKS Jianing Song, Table of n, a(n) for n = 0..10000 N. J. A. Sloane et al., Binary Quadratic Forms and OEIS. FORMULA G.f.: 1 + 2 * Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k). a(n) = 2 * A318983(n) unless n = 0. a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(163^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, b(p^e) = e + 1 if Kronecker(-163, p) = 1. Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(163) = 0.49213705... . - Amiram Eldar, Dec 16 2023 EXAMPLE G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + 4*x^41 + 4*x^43 + 4*x^47 + 2*x^49 + 4*x^53 + 4*x^61 + 2*x^64 + 4*x^71 + ... MATHEMATICA Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-163, #] &]]; 2 Table[a[n], {n, 1, 110}]] (* Vincenzo Librandi, Sep 10 2018 *) PROG (PARI) a(n) = if(n, 2*sumdiv(n, d, kronecker(-163, d)), 1) CROSSREFS Cf. A318983. 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), A318984 (d=-67), this sequence (d=-163). Sequence in context: A033759 A033755 A033753 * A033749 A033743 A260676 Adjacent sequences: A318982 A318983 A318984 * A318986 A318987 A318988 KEYWORD nonn,easy AUTHOR Jianing Song, Sep 06 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 20 23:09 EDT 2024. Contains 372720 sequences. (Running on oeis4.)