A140966 a(n) = (5 + (-2)^n)/3.

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Author

Paul Curtz, Jul 27 2008

Keywords

• sign
• easy

Comments

Inverse binomial transform of A048573.

This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).

The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.

Sequences with k=2 are A094554 and A094555.

Sequences with k=3 are A084175, A108924, and A139818.

Links

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

Crossrefs

Cf. A048573.

Cf. A097163, A135520, A136326, A136336, A137208.

Cf. A094554, A094555.

Cf. A084175, A108924, A139818.

Cf. A000079, A001045, A078008, A083582, A083584, A163834.

Programs

Magma

[( 5+(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

Mathematica

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

PARI

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = -a(n-1) + 2*a(n-2).

G.f.: (2+3*x)/((1-x)*(1+2*x)).

a(n+1) - a(n) = (-1)^(n+1)*A000079(n).

a(n+3) = (-1)^n*A083582(n).

a(n+1) - 2*a(n) = -a(n+2).

a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).

a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014

E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Extensions

Definition simplified by R. J. Mathar, Sep 11 2009