A141576 a(0)=-1, a(1)=0, a(2)=1, a(n) = a(n-1) - 2*a(n-2) + a(n-3).

-1, 0, 1, 0, -2, -1, 3, 3, -4, -7, 4, 14, -1, -25, -9, 40, 33, -56, -82, 63, 171, -37, -316, -71, 524, 350, -769, -945, 943, 2064, -767, -3952, -354, 6783, 3539, -10381, -10676, 13625, 24596, -13330, -48897, 2359, 86823, 33208, -138079, -117672, 191694

Offset

0,5

References

Martin Gardner, Mathematical Circus, Random House, New York, 1981, p. 165.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,-2,1).

Formula

From R. J. Mathar, Aug 25 2008: (Start)

O.g.f.: -(1-x+x^2)/(1-x+2*x^2-x^3).

a(n) = A078019(n-2), n > 0. (End)

a(n) = -A000931(-2*n + 3). - Michael Somos, Sep 18 2012

Example

G.f. = -1 + x^2 - 2*x^4 - x^5 + 3*x^6 + 3*x^7 - 4*x^8 - 7*x^9 + ...

Mathematica

Nest[Append[#, #[[-1]] - 2 #[[-2]] + #[[-3]]] &, {-1, 0, 1}, 44] (* Michael De Vlieger, Dec 17 2017 *)
LinearRecurrence[{1, -2, 1}, {-1, 0, 1}, 50] (* Harvey P. Dale, Feb 06 2024 *)

Prog

(MATLAB)
function y=fib(n)
%Generates difference sequence
fz(1)=-1; fz(2)=0; fz(3)=1;
for k=4:n
fz(k)=fz(k-1)-2*fz(k-2)+fz(k-3);
end
y=fz(n);
(PARI) x='x+O('x^99); Vec((1-x+x^2)/(-1+x-2*x^2+x^3)) \\ Altug Alkan, Dec 17 2017
(Magma)
I:=[-1, 0, 1]; [n le 3 select I[n] else Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..50]]; // G. C. Greubel, Sep 16 2024
(SageMath)
def A141576_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( -(1-x+x^2)/(1-x+2*x^2-x^3) ).list()
A141576_list(50) # G. C. Greubel, Sep 16 2024

Crossrefs

Cf. A000931, A078019, A141577.

Sequence in context: A117363 A007307 A207617 * A078019 A038071 A032140

Adjacent sequences: A141573 A141574 A141575 * A141577 A141578 A141579

Keyword

sign,easy

Author

Matt Wynne (mattwyn(AT)verizon.net), Aug 18 2008

Extensions

Extended by R. J. Mathar, Aug 25 2008

Status

approved