A114285 Expansion of g.f. (1 - 3*x)/((1 - x)*(1 - x^2)).

1, -2, -1, -4, -3, -6, -5, -8, -7, -10, -9, -12, -11, -14, -13, -16, -15, -18, -17, -20, -19, -22, -21, -24, -23, -26, -25, -28, -27, -30, -29, -32, -31, -34, -33, -36, -35, -38, -37, -40, -39, -42, -41, -44, -43, -46, -45, -48, -47, -50, -49, -52, -51, -54, -53, -56, -55, -58, -57, -60, -59, -62, -61

Offset: 0

Author

Paul Barry, Nov 20 2005

Keywords

• easy
• sign

Comments

Diagonal sums of A114284.

Links

• Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A103889, A114284.

Programs

Magma

[(-1)^n-n : n in [0..70]]; // Wesley Ivan Hurt, Sep 06 2015

Magma

I:=[1,-2,-1]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..70]]; // Vincenzo Librandi, Sep 07 2015

Maple

A114285:=n->(-1)^n-n: seq(A114285(n), n=0..70); # Wesley Ivan Hurt, Sep 06 2015

Mathematica

Table[(-1)^n-n, {n,0,70}] (* Wesley Ivan Hurt, Sep 06 2015 *)
CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - x^2)), {x, 0, 70}] ,x] (* Vincenzo Librandi, Sep 07 2015 *)
LinearRecurrence[{1,1,-1},{1,-2,-1},70] (* Harvey P. Dale, Jul 24 2019 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} 3*0^(n-2k) - 2.

a(n) = 3*(1 + (-1)^n)/2 - 2*floor((n+2)/2).

a(n) = - A103889(n). - R. J. Mathar, Apr 06 2008

From Wesley Ivan Hurt, Sep 06 2015: (Start)

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

a(n) = (-1)^n - n. (End)

E.g.f.: exp(-x) - x*exp(x). - Stefano Spezia, May 03 2023