OFFSET
0,1
COMMENTS
The sequence is the non-principal Dirichlet character of the reduced residue system mod 3. (The other is A011655.) Associated Dirichlet L-functions are L(1, chi) = Sum_{n >= 1} a(n)/n = A073010, L(2, chi)= Sum_{n >= 1} a(n)/n^2 = A086724, or L(3, chi)= Sum_{n >= 1} a(n)/n^3 = A129404. [Jolley eq 310] - R. J. Mathar, Jul 15 2010
a(n) = 2*D(n) - L(n), where L(n) denotes the n-th Lucas number and D(n) denotes the so-called n-th quadrapell number -- defined and discussed by Dursun Tasci in his paper (see References below). We have D(n) = D(n-2) + 2*D(n-3) + D(n-4), D(0) = D(1) = D(2) = 1, D(3) = 2. G.f. D(x) = (1+x-x^3)/((1-x-x^2)(1+x+x^2)). - Roman Witula, Jul 31 2012
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = 0, z = -1. - Michael Somos, Nov 27 2019
Second highest lowest zero among degree 1 L-functions [LMFDB]. - Michael Somos, Sep 23 2025
The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-3)). - Jianing Song, Dec 13 2025
REFERENCES
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
L. B. W. Jolley, Summation of Series, Dover Publications, 1961.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 6.
LINKS
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
LMFDB, L-function 1-3-3.2-r1-0-0.
R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015, Table 2, Table 22 for m=3, r=2.
Dursun Tasci, On Quadrapell Numbers and Quadrapell Polynomials, Hacettepe J. Math. Stat., 38 (3) (2009), 265-275.
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
FORMULA
a(n) = A049347(n-1).
a(n) = -a(n-1) - a(n-2); a(0) = 0, a(1) = 1. G.f.: x/(1+x+x^2). - Philippe Deléham, Nov 03 2008
a(n) = -2*sin(4*Pi*n/3)/sqrt(3) = 2*sin(8*Pi*n/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008
a(n) = 2*sin(2*Pi*n/3)/sqrt(3). - Roman Witula, Jul 31 2012
a(n) = Legendre(n, 3), the Legendre symbol for p = 3. - Alonso del Arte, Feb 06 2013
a(n) = (-3/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. - Wolfdieter Lang, May 29 2013
Dirichlet g.f.: L(chi_2(3),s), with chi_2(3) the nontrivial Dirichlet character modulo 3. - Ralf Stephan, Mar 27 2015
a(n) = a(n-3) for n > 2. - Wesley Ivan Hurt, Jul 02 2016
E.g.f.: 2*sin(sqrt(3)*x/2)*exp(-x/2)/sqrt(3). - Ilya Gutkovskiy, Jul 02 2016
a(n) = H(2*n, 1, 1/2) for n > 0 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 4). - Peter Luschny, Sep 03 2019
Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Nov 27 2019
a(n) = n - 3*floor((n+1)/3). - Wolfdieter Lang, Oct 07 2021
Sum_{n > 0} a(n)*x^(2*n)/(2*n)! = 2/sqrt(3) * sin(sqrt(3)*x/2) * sinh (x/2) . - Michael Somos, Sep 23 2025
Sum_{n>=1} a(n)/n = -(Pi/3^(3/2)) * (Sum_{i=0..2} i*a(i)) = Pi/(3*sqrt(3)) (Dirichlet class number formula). - Jianing Song, Dec 13 2025
Completely multiplicative with a(3) = 0, a(p) = 1 if p == 1 (mod 3), and a(p) = -1 if p == 2 (mod 3). - Amiram Eldar, May 23 2026
EXAMPLE
G.f. = x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + ... - Michael Somos, Nov 27 2019
Dirichlet series L(s) = 1 - 2^-s + 4^-s - 5^-5 + 7^-s - 8^-s + 10^-2 + .... - Michael Somos, Sep 23 2025
MAPLE
ch:=n-> if n mod 3 = 0 then 0; elif n mod 3 = 1 then 1; else -1; fi;
seq(op([0, 1, -1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016
MATHEMATICA
Table[JacobiSymbol[n, 3], {n, 0, 99}] (* Alonso del Arte, Feb 06 2013 *)
(* Alternative: *)
Table[KroneckerSymbol[-3, n], {n, 0, 99}] (* Wolfdieter Lang, May 30 2013 *)
(* Alternative: *)
PadRight[{}, 100, {0, 1, -1}] (* Wesley Ivan Hurt, Jul 02 2016 *)
(* Alternative: *)
a[ n_] := {1, -1, 0}[[Mod[n, 3, 1]]]; (* Michael Somos, Nov 27 2019 *)
PROG
(SageMath)
def A102283():
x, y = 0, -1
while True:
yield -x
x, y = y, -x -y
a = A102283(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013
(Magma) &cat [[0, 1, -1]^^30]; // Wesley Ivan Hurt, Jul 02 2016
(PARI) a(n)=([0, 1; -1, -1]^n*[0; 1])[1, 1] \\ Charles R Greathouse IV, Jan 14 2017
(PARI) {a(n) = [0, 1, -1][n%3 + 1]}; /* Michael Somos, Nov 27 2019 */
(Python)
def A102283(n): return (0, 1, -1)[n%3] # Chai Wah Wu, Sep 16 2023
CROSSREFS
Moebius transform of A002324.
Cf. A007645 (primes not inert in Q(sqrt(-3))), A002476 (primes decomposing), A003627 (primes remaining inert), A045309 (primes not decomposing).
Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: this sequence, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.
KEYWORD
sign,easy,mult
AUTHOR
N. J. A. Sloane, Nov 02 2008
STATUS
approved