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A102283 Period 3: repeat [0, 1, -1]. 19
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 (list; graph; refs; listen; history; text; internal format)
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) denote the n-th Lucas number, whereas D(n) denote, 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

REFERENCES

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.

L. B. W. Jolley, Summation of Series, Dover (1961).

LINKS

Table of n, a(n) for n=0..104.

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).

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) = -(1/3)*{(n mod 3) - 2*[(n+1) mod 3] + [(n+2) mod 3]}. a(n) = -(1/3)*I*sqrt(3)*[ -1/2+(1/2)*I*sqrt(3)]^n + (1/3)*I*sqrt(3)*[ -1/2 - (1/2)*I *sqrt(3)]^n, with n >= 0 and I = sqrt(-1). - Paolo P. Lava, Nov 06 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), which immediately follows from Paolo Lava's formula. - 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

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 *)

PROG

(Sage)

def A102283():

    x, y = 0, -1

    while true:

        yield -x

        x, y = y, -x -y

a = A102283(); [a.next() 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

CROSSREFS

Cf. A011655, A049347, A073010, A086724, A129404.

Sequence in context: A092220 A011655 * A128834 A022928 A000494 A022933

Adjacent sequences:  A102280 A102281 A102282 * A102284 A102285 A102286

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane, Nov 02 2008

STATUS

approved

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Last modified August 21 01:18 EDT 2017. Contains 290855 sequences.