A075193 Expansion of (1-2*x)/(1+x-x^2).

1, -3, 4, -7, 11, -18, 29, -47, 76, -123, 199, -322, 521, -843, 1364, -2207, 3571, -5778, 9349, -15127, 24476, -39603, 64079, -103682, 167761, -271443, 439204, -710647, 1149851, -1860498, 3010349, -4870847, 7881196, -12752043, 20633239, -33385282, 54018521, -87403803, 141422324

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Sep 07 2002

Keywords

• easy
• sign

Comments

"Inverted" Lucas numbers:

The g.f. is obtained inserting 1/x into the g.f. of Lucas sequence and dividing by x. The closed form is a(n)=(-1)^n*a^(n+1)+(-1)^n*b^(n+1), where a=golden ratio and b=1-a, so that a(n)=(-1)^n*L(n+1), L(n)=Lucas numbers.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (-1,1).

Crossrefs

Cf. A000032, A000045, A001906, A039834, A186679.

Programs

Haskell

a075193 n = a075193_list !! n
a075193_list = 1 : -3 : zipWith (-) a075193_list (tail a075193_list)
-- Reinhard Zumkeller, Sep 15 2015

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-2*x)/(1+x-x^2))); // Marius A. Burtea, Jan 12 2020

Maple

a:= n-> (Matrix([[1, -2]]). Matrix([[-1, 1], [1, 0]])^(n))[1, 1]:
seq(a(n), n=0..45); # Alois P. Heinz, Jul 31 2008

Mathematica

CoefficientList[Series[(1 - 2z)/(1 + z - z^2), {z, 0, 40}], z]

Formula

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

a(n) = term (1,1) in the 1x2 matrix [1,-2] * [-1,1; 1,0]^n. - Alois P. Heinz, Jul 31 2008

a(n) = A186679(n)+A186679(n-2) for n>1. - Reinhard Zumkeller, Feb 25 2011

a(n) = A039834(n+1)-2*A039834(n). - R. J. Mathar, Sep 27 2014

a(n) = (-1)^(n-1)*A001906(n)/A000045(n). - Taras Goy, Jan 12 2020

E.g.f.: exp(-(1+sqrt(5))*x/2)*(3 + sqrt(5) - 2*exp(sqrt(5)*x))/(1 + sqrt(5)). - Stefano Spezia, Jan 12 2020