A175629 Legendre symbol (n,7).

0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1

Offset: 0

Author

R. J. Mathar, Jul 29 2010

Keywords

• easy
• mult
• sign

Comments

This represents a non-principal Dirichlet character modulo 7.

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=7, Chi_2(n).

Links

• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1).

Crossrefs

Cf. A089509, A097343.

The Legendre symbols (n,p): A091337 (p = 2, Kronecker symbol), A102283 (p = 3), A080891 (p = 5), this sequence (p = 7), A011582 (p = 11), A011583 (p = 13), ..., A011631 (p = 251), A165573 (p = 257), A165574 (p = 263). Also, many other sequences for p > 263 are in the OEIS.

Moebius transform of A035182.

Programs

Magma

&cat [[0, 1, 1, -1, 1, -1, -1]^^20]; // Vincenzo Librandi, Jun 30 2018

Maple

A := proc(n) numtheory[jacobi](n,7) ; end proc: seq(A(n),n=0..120) ;

Mathematica

LinearRecurrence[{-1,-1,-1,-1,-1,-1},{0,1,1,-1,1,-1},100] (* or *) PadRight[ {},100,{0,1,1,-1,1,-1,-1}] (* Harvey P. Dale, Aug 02 2013 *)
Table[JacobiSymbol[n, 7], {n, 0, 100}] (* Vincenzo Librandi, Jun 30 2018 *)

PARI

a(n) = kronecker(n, 7); \\ Michel Marcus, Jan 28 2019

Formula

a(n) = a(n+7).

|a(n)| = A109720(n).

a(n) = -a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - a(n-6).

G.f.: x*(1 + 2*x + x^2 + 2*x^3 + x^4)/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6).

a(n) == n^3 (mod 7). - Jianing Song, Jun 29 2018