A397518 a(n) is the number of integer solutions of x^2 + 7*y^2 = 2*n + 1.

2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 4, 2, 0, 4, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 0, 0, 0, 2, 0, 4, 0, 4, 0, 0, 4, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 4, 0, 4, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0

Offset

0,1

Links

Table of n, a(n) for n=0..99.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, Exercise n. 5 at p. 346.

Formula

a(n) = 2*Sum_{d | 2*n+1} JacobiSymbol(-7,d) (see Uspensky and Heaslet).

Example

a(0) = 2 since 2*0 + 1 = 1  equals 1^2 + 7*0^2 and (-1)^2 + 7*0^2;
a(3) = 2 since 2*3 + 1 = 7 equals 0^2 + 7*1^2 and 0^2 + 7*(-1)^2;
a(5) = 4 since 2*5 + 1 = 11 equals 2^2 + 7*1^2, (-2)^2 + 7*1^2, 2^2 + 7*(-1)^2, and (-2)^2 + 7*(-1)^2.

Mathematica

a[n_]:=2*Sum[JacobiSymbol[-7, d], {d, Divisors[2n+1]}]; Array[a, 100, 0]

Crossrefs

Cf. A005881, A274658.

Sequence in context: A262904 A390524 A391449 * A144377 A138527 A033718

Adjacent sequences: A397514 A397515 A397516 * A397519 A397520 A397521

Keyword

nonn

Author

Stefano Spezia, Jun 29 2026

Status

approved