A316569 a(n) = Jacobi (or Kronecker) symbol (n, 15).

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

Offset: 0

Author

Jianing Song, Aug 05 2018

Keywords

• sign
• easy
• mult

Comments

Period 15: repeat [0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1].

Also a(n) = Kronecker(-15, n).

This sequence is one of the three non-principal real Dirichlet characters modulo 15. The other two are Jacobi or Kronecker symbols (n, 45) (or (45, n)) and (n, 75) (or (-75, n)).

Note that (Sum_{i=0..14} i*a(i))/(-15) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-15)).

Links

• Eric Weisstein's World of Mathematics, Kronecker Symbol
• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,-1,0,1,-1).

Crossrefs

Cf. A035175 (inverse Moebius transform).

Kronecker symbols: A063524 ((n, 0) or (0, n)), A000012 ((n, 1) or (1, n)), A091337 ((n, 2) or (2, n) or (n, 8) or (8, n)), A102283 ((n, 3) or (-3, n)), A000035 ((n, 4) or (4, n) or (n, 16) or (16, n)), A080891 ((n, 5) or (5, n)), A109017 ((n, 6) or (-6, n)), A175629 ((n, 7) or (-7, n)), A011655 ((n, 9) or (9, n)), A011582 ((n, 11) or (-11, n)), A134667 ((n, 12) or (-12, n)), A011583 ((n, 13) or (13, n)), this sequence ((n, 15) or (-15, n)).

Programs

Magma

[KroneckerSymbol(-15, n): n in [0..100]]; // Vincenzo Librandi, Aug 28 2018

Mathematica

Array[ JacobiSymbol[#, 15] &, 90, 0] (* Robert G. Wilson v, Aug 06 2018 *)
PadRight[{},100,{0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1}] (* Harvey P. Dale, Feb 20 2023 *)

PARI

a(n) = kronecker(n, 15)

Formula

a(n) = 1 for n == 1, 2, 4, 8 (mod 15); -1 for n == 7, 11, 13, 14 (mod 15); 0 for n that are not coprime with 15.

Completely multiplicative with a(p) = a(p mod 15) for primes p.

a(n) = A102283(n)*A080891(n).

a(n) = a(n+15) = -a(-n) for all n in Z.

From Chai Wah Wu, Feb 16 2021: (Start)

a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-5) + a(n-7) - a(n-8) for n > 7.

G.f.: (x^7 - x^5 + 2*x^4 - x^3 + x)/(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1). (End)