|
|
A102283
|
|
Period 3: repeat [0, 1, -1].
|
|
46
|
|
|
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) 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
|
|
REFERENCES
|
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
L. B. W. Jolley, Summation of Series, Dover Publications (1961).
|
|
LINKS
|
|
|
FORMULA
|
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) = 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
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
|
|
EXAMPLE
|
G.f. = x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + ... - Michael Somos, Nov 27 2019
|
|
MAPLE
|
ch:=n-> if n mod 3 = 0 then 0; elif n mod 3 = 1 then 1; else -1; fi;
|
|
MATHEMATICA
|
Table[KroneckerSymbol[-3, n], {n, 0, 99}] (* Wolfdieter Lang, May 30 2013 *)
a[ n_] := {1, -1, 0}[[Mod[n, 3, 1]]]; (* Michael Somos, Nov 27 2019 *)
|
|
PROG
|
(Sage)
x, y = 0, -1
while True:
yield -x
x, y = y, -x -y
(PARI) {a(n) = [0, 1, -1][n%3 + 1]}; /* Michael Somos, Nov 27 2019 */
(Python)
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|