A130707 a(n+3) = 3*(a(n+2) - a(n+1)) + 2*a(n).

1, 2, 2, 2, 4, 10, 22, 44, 86, 170, 340, 682, 1366, 2732, 5462, 10922, 21844, 43690, 87382, 174764, 349526, 699050, 1398100, 2796202, 5592406, 11184812, 22369622, 44739242, 89478484, 178956970, 357913942, 715827884, 1431655766, 2863311530

Offset

0,2

Comments

Binomial transform of period-3 sequence with period 1 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).

Formula

a(n) = 2^n/3 + 4*(-1)^n*(1/3)*cos((2n+1)*Pi/3). - Emeric Deutsch, Jul 27 2007

From R. J. Mathar, Nov 18 2007: (Start)

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

a(n) = (2*A057079(n) + 2^n)/3. (End)

Maple

a:=proc(n) options operator, arrow: (1/3)*2^n+(4/3)*(-1)^n*cos((1/3)*(2*n+1)*Pi) end proc: seq(a(n), n = 0 .. 33); # Emeric Deutsch, Jul 27 2007

Mathematica

RecurrenceTable[{a[0]==1, a[1]==a[2]==2, a[n]==3(a[n-1]-a[n-2])+2a[n-3]}, a, {n, 40}] (* Harvey P. Dale, Jan 18 2015 *)
(* Alternative: *)
LinearRecurrence[{3, -3, 2}, {1, 2, 2}, 40] (* Harvey P. Dale, Jan 18 2015 *)

Prog

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 2, -3, 3]^n*[1; 2; 2])[1, 1] \\ Charles R Greathouse IV, May 27 2026

Crossrefs

Cf. A057079.

Sequence in context: A360314 A213270 A307522 * A131562 A260786 A374663

Adjacent sequences: A130704 A130705 A130706 * A130708 A130709 A130710

Keyword

nonn,easy,changed

Author

Paul Curtz, Jul 01 2007

Extensions

More terms from Emeric Deutsch, Jul 27 2007

Status

approved