OFFSET
1,3
COMMENTS
WARNING: It is unclear whether this sequence should start at offset 1 (as written) or offset 0 (in analogy to many similar sequences, which seems to be assumed in many of the given formulas).
Inverse binomial transform of A057079. - Paul Barry, May 15 2003
The unsigned version, with g.f. (1 + x + 2*x^2)/(1 - x^3), has a(n) = 4/3 -cos(2*Pi*n/3)/3 - sqrt(3)*sin(2*Pi*n/3)/3 = gcd(Fib(n+4), Fib(n+1)). - Paul Barry, Apr 02 2004
a(n) = L(n-2,-1), where L is defined as in A108299; see also A010892 for L(n,+1). - Reinhard Zumkeller, Jun 01 2005
From the Taylor expansion of log(1 + x + x^2) at x = 1, Sum_{k > 0} a(k)/k = log(3) = A002391. This is case n = 3 of the general expression Sum_{k > 0} (1-n*!(k mod n))/k = log(n). - Jaume Oliver Lafont, Oct 16 2009
If used with offset zero, a non-simple continued fraction representation of 2+sqrt(2). - R. J. Mathar, Mar 08 2012
Periodic sequences of this type can be also calculated by a(n) = c + floor(q/(p^m-1)*p^n) mod p, where c is a constant, q is the number representing the periodic digit pattern and m is the period length. c, p and q can be calculated as follows: Let D be the array representing the number pattern to be repeated, m = size of D, max = maximum value of elements in D, min = minimum value of elements in D. Than c := min, p := max - min + 1 and q := p^m*Sum_{i=1..m} (D(i)-min)/p^i. Example: D = (1, 1, -2), c = -2, p = 4 and q = 60 for this sequence. - Hieronymus Fischer, Jan 04 2013
This is the Dirichlet inverse of A117997. - Petros Hadjicostas, Jul 25 2020
LINKS
W. Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Ralph E. Griswold, Shaft Sequences, 2001 (see also here).
Tanya Khovanova, Recursive Sequences.
W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31(1) (1993), 2-6.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
FORMULA
With offset zero, a(n) = A057079(2n). a(n) = -a(n-1) - a(n-2) with a(0) = a(1) = 1.
From Mario Catalani (mario.catalani(AT)unito.it), Jan 07 2003: (Start)
G.f.: x*(1 + 2*x)/(1 + x + x^2).
a(n) = (-1)^floor(2n/3) + ((-1)^floor((2n-1)/3) + (-1)^floor((2n+1)/3))/2. (End)
a(n) = -2*cos(2*Pi*n/3). - Jaume Oliver Lafont, May 06 2008
Dirichlet g.f.: zeta(s)*(1-1/3^(s-1)). - R. J. Mathar, Feb 09 2011
a(n) = n * Sum_{k=1..n} binomial(k,n-k)/k*(-1)^(k+1). - Dmitry Kruchinin, Jun 03 2011
a(n) = -2 + floor(110/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013
a(n) = -2 + floor(20/21*4^(n+1)) mod 4. - Hieronymus Fischer, Jan 04 2013
a(n) = a(n-3) for n > 3. - Wesley Ivan Hurt, Jul 01 2016
E.g.f.: 2 - 2*cos(sqrt(3)*x/2)*exp(-x/2). - Ilya Gutkovskiy, Jul 01 2016
a(n) = (-1)^n*hypergeom([-n/2-1, -n/2-3/2], [-n-2], 4). - Peter Luschny, Dec 17 2016
EXAMPLE
G.f.: x + x^2 - 2*x^3 + x^4 + x^5 - 2*x^6 + x^7 + x^8 - 2*x^9 + ... - Michael Somos, Nov 27 2019
MAPLE
seq(op([1, 1, -2]), n=1..50); # Wesley Ivan Hurt, Jul 01 2016
MATHEMATICA
a[n_] := {1, 1, -2}[[Mod[n - 1, 3] + 1]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{}, 90, {1, 1, -2}] (* After Harvey P. Dale, or *)
CoefficientList[ Series[(2x + 1)/(x^2 + x + 1), {x, 0, 89}], x] (* or *)
LinearRecurrence[{-1, -1}, {1, 1}, 90] (* Robert G. Wilson v, Jul 30 2018 *)
PROG
(PARI) a(n)=1-3*!(n%3) \\ Jaume Oliver Lafont, Oct 16 2009
(Sage)
def A061347():
x, y = -1, -1
while True:
yield -x
x, y = y, -x -y
a = A061347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013
(Magma) &cat [[1, 1, -2]^^30]; // Wesley Ivan Hurt, Jul 01 2016
(GAP) Flat(List([1..50], n->[1, 1, -2])); # Muniru A Asiru, Aug 02 2018
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Jason Earls, Jun 07 2001
EXTENSIONS
Better definition from M. F. Hasler, Jan 13 2013
STATUS
approved