A102283 Period 3: repeat [0, 1, -1].

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

Offset: 0

Author

N. J. A. Sloane, Nov 02 2008

Keywords

• sign
• easy
• mult

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

Examples

			G.f. = x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + ... - _Michael Somos_, Nov 27 2019
		

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.
• R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015, Table 2, Table 22 for m=3, r=2.
• D. 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).

Crossrefs

Cf. A011655, A049347, A073010, A086724, A129404, A002324 (Mobius transform).

Programs

Magma

&cat [[0, 1, -1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

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 *)
Table[KroneckerSymbol[-3, n], {n, 0, 99}] (* Wolfdieter Lang, May 30 2013 *)
PadRight[{}, 100, {0, 1, -1}] (* Wesley Ivan Hurt, Jul 02 2016 *)
a[ n_] := {1, -1, 0}[[Mod[n, 3, 1]]]; (* Michael Somos, Nov 27 2019 *)

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

Sage

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

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