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
Examples
G.f. = -1 + x^2 - 2*x^4 - x^5 + 3*x^6 + 3*x^7 - 4*x^8 - 7*x^9 + ...
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).
Programs
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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);
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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
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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 *) -
PARI
x='x+O('x^99); Vec((1-x+x^2)/(-1+x-2*x^2+x^3)) \\ Altug Alkan, Dec 17 2017 -
SageMath
def A141576_list(prec): P.
= 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
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
Extensions
Extended by R. J. Mathar, Aug 25 2008