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%I #50 Sep 08 2022 08:45:51
%S 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,
%T 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,
%U 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
%N Legendre symbol (n,7).
%C This represents a non-principal Dirichlet character modulo 7.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=7, Chi_2(n).
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1,-1,-1).
%F a(n) = a(n+7).
%F |a(n)| = A109720(n).
%F a(n) = -a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - a(n-6).
%F 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).
%F a(n) == n^3 (mod 7). - _Jianing Song_, Jun 29 2018
%p A := proc(n) numtheory[jacobi](n,7) ; end proc: seq(A(n),n=0..120) ;
%t 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 *)
%t Table[JacobiSymbol[n, 7], {n, 0, 100}] (* _Vincenzo Librandi_, Jun 30 2018 *)
%o (Magma) &cat [[0, 1, 1, -1, 1, -1, -1]^^20]; // _Vincenzo Librandi_, Jun 30 2018
%o (PARI) a(n) = kronecker(n, 7); \\ _Michel Marcus_, Jan 28 2019
%Y Cf. A089509, A097343.
%Y 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.
%Y Moebius transform of A035182.
%K easy,mult,sign
%O 0,1
%A _R. J. Mathar_, Jul 29 2010
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