A142585 Inverse binomial transform of A014217.

1, 0, 1, 0, -1, 5, -14, 35, -81, 180, -389, 825, -1726, 3575, -7349, 15020, -30561, 61965, -125294, 252795, -509161, 1024100, -2057549, 4130225, -8284926, 16609455, -33282989, 66669660, -133507081, 267285605, -535010414, 1070731475, -2142612801, 4287086100

Offset: 0

Author

Paul Curtz, Sep 21 2008

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-3,-1,2).

Crossrefs

Cf. A000032, A000045, A014217, A027974.

Programs

Magma

[1] cat [(-1)^n*( Lucas(n) - 2^(n-1) ): n in [1..40]]; // G. C. Greubel, Apr 14 2021

Mathematica

Table[(-1)^n*(LucasL[n] -2^(n-1)) - Boole[n==0]/2, {n,0,40}] (* G. C. Greubel, Apr 14 2021 *)

Sage

[1]+[(-1)^n*( lucas_number2(n,1,-1) - 2^(n-1) ) for n in (1..40)] # G. C. Greubel, Apr 14 2021

Formula

a(n) = (-1)^(n+1) * A027974(n-4) for n > 4.

G.f.: (1+3*x+2*x^2+x^3)/((1+2*x)*(1+x-x^2)). - R. J. Mathar, Sep 22 2008

a(n) = (-1)^(n+1)*(2^(n-1) -F(n+1) -F(n-1)), where F(n) = A000045(n), for n>=1, with a(0)=1. - Johannes W. Meijer, Aug 15 2010

a(n) = -3*a(n-1) - a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Oct 06 2017

Extensions

Edited and extended by R. J. Mathar, Sep 22 2008