A131370 a(n) = 3a(n-1) - 3a(n-2) + 2a(n-3), a(0) = 3, a(1) = 2, a(2) = 0.

3, 2, 0, 0, 4, 12, 24, 44, 84, 168, 340, 684, 1368, 2732, 5460, 10920, 21844, 43692, 87384, 174764, 349524, 699048, 1398100, 2796204, 5592408, 11184812, 22369620, 44739240, 89478484, 178956972, 357913944, 715827884, 1431655764, 2863311528

Offset: 0

Author

Paul Curtz, Sep 30 2007

Keywords

• nonn
• easy

Comments

Sequence is identical to its third differences. Binomial transform of 3, -1, -1, 3, -1, -1, 3, -1, -1, ... .

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3, -3, 2).

Crossrefs

Cf. A086953.

Programs

Maple

seq((1/3)*2^n+8*cos((1/3)*n*Pi)*1/3,n=0..33); # Emeric Deutsch, Oct 15 2007

Mathematica

a = {3, 2, 0}; Do[AppendTo[a, 3*a[[ -1]] - 3*a[[ -2]] + 2*a[[ -3]]], {60}]; a (* Stefan Steinerberger, Oct 04 2007 *)
LinearRecurrence[{3,-3,2},{3,2,0},40] (* Harvey P. Dale, Apr 28 2025 *)

Formula

a(n) = 2^n/3 + (8/3)cos(n*Pi/3). - Emeric Deutsch, Oct 15 2007

G.f.: -(3-7*x+3*x^2)/(2*x-1)/(x^2-x+1). - R. J. Mathar, Nov 14 2007

a(n) = 2*A086953(n-1) for n>0. - Rick L. Shepherd, Aug 02 2017

Extensions

More terms from Stefan Steinerberger, Oct 04 2007