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

Author

Jianing Song, Dec 27 2018

Keywords

• sign
• easy
• mult

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

• 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).

Crossrefs

Cf. A035203 (inverse Moebius transform).

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017(d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), this sequence (d=21), A322796 (d=24).

Programs

Mathematica

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

PARI

a(n) = kronecker(n, 21)

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)