A083944 A generalized Jacobsthal sequence.

0, 1, -2, -3, -10, -19, -42, -83, -170, -339, -682, -1363, -2730, -5459, -10922, -21843, -43690, -87379, -174762, -349523, -699050, -1398099, -2796202, -5592403, -11184810, -22369619, -44739242, -89478483, -178956970, -357913939, -715827882, -1431655763

Offset: 0

Author

Paul Barry, May 09 2003

Keywords

• sign
• easy

Links

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

Crossrefs

Cf. A083943.

Programs

Magma

[3/2-2^(n+1)/3-5*(-1)^n/6: n in [0..40]]; // Vincenzo Librandi, Apr 04 2012

Mathematica

CoefficientList[Series[x (1-4x)/((1+x)(1-x)(1-2x)),{x,0,40}],x] (* Vincenzo Librandi, Apr 04 2012 *)
LinearRecurrence[{2,1,-2},{0,1,-2},40] (* Harvey P. Dale, Jun 08 2014 *)

PARI

x='x+O('x^50); concat([0], Vec(x*(1-4*x)/((1+x)*(1-x)*(1-2*x)))) \\ G. C. Greubel, Oct 10 2017

Formula

G.f.: x*(1-4*x)/((1+x)*(1-x)*(1-2*x)).

E.g.f.: (9*exp(x) - 4*exp(2*x) - 5*exp(-x))/6.

a(n) = (9 - 2^(n+2) - 5*(-1)^n)/6.

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

a(2*n) = -A000975(2*n); a(2*n+1) = 2 - A000975(2*n+1).

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), with a(0)=0, a(1)=1, a(2)=-2. - Harvey P. Dale, Jun 08 2014