A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

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

Offset

0,1

Comments

Period 21: repeat [0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1].

Also a(n) = Kronecker symbol (21/n).

This sequence is one of the three non-principal real Dirichlet characters modulo 21. The other two are Jacobi or Kronecker symbols {(n/63)} (or {(-63/n)}) and {(n/147)} (or {(-147/n)}).

Links

Table of n, a(n) for n=0..84.

Eric Weisstein's World of Mathematics, Kronecker Symbol

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,0,1,-1,0,1,-1).

Formula

a(n) = 1 for n == 1, 4, 5, 16, 17, 20 (mod 21); -1 for n == 2, 8, 10, 11, 13, 19 (mod 21); 0 for n that are not coprime with 21.

Completely multiplicative with a(p) = a(p mod 21) for primes p.

a(n) = A102283(n)*A175629(n).

a(n) = a(n+21) = -a(n) for all n in Z.

From Chai Wah Wu, Feb 18 2021: (Start)

a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-8) - a(n-9) + a(n-11) - a(n-12) for n > 11.

G.f.: -x*(x - 1)*(x + 1)*(x^8 - 2*x^7 + 2*x^6 + 2*x^2 - 2*x + 1)/(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1). (End)

Mathematica

JacobiSymbol[Range[0, 100], 21] (* Paolo Xausa, Mar 19 2025 *)

Prog

(PARI) a(n) = kronecker(n, 21)

Crossrefs

Moebius transform of A035203.

Cf. A038893 (primes not inert in Q(sqrt(21))), A038894 (primes remaining inert).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.

Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: A080891, A091337, A110161, A011583, A011584, this sequence, A322796, A390615, A011587, A391502, A011589, A391503, A011590.

Sequence in context: A380172 A171587 A284627 * A286400 A288478 A225183

Adjacent sequences: A322826 A322827 A322828 * A322830 A322831 A322832

Keyword

sign,easy,mult

Author

Jianing Song, Dec 27 2018

Status

approved