A166022 a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +3*a(n-4) -2*a(n-5) for n > 4, with initial values as shown.

0, 1, 2, 2, 4, 9, 21, 43, 86, 170, 340, 681, 1365, 2731, 5462, 10922, 21844, 43689, 87381, 174763, 349526, 699050, 1398100, 2796201, 5592405, 11184811, 22369622, 44739242, 89478484, 178956969, 357913941, 715827883, 1431655766, 2863311530, 5726623060, 11453246121

Offset

0,3

Comments

a(n+1)-2*a(n) defines a periodic sequence with period length 6: 1, 0, -2, 0, 1, 3 (repeat).

The sequence a(n) and its iterated differences in successive lines are:

..0, 1, .2, .2, .4, .9, 21, 43, 86, 170, 340,...

..1, 1, .0, .2, .5, 12, 22, 43, 84, 170, 341,...

..0,-1, .2, .3, .7, 10, 21, 41, 86, 171, 343,...

.-1, 3, .1, .4, .3, 11, 20, 45, 85, 172, 339,...

..4,-2, .3, -1, .8, .9, 25, 40, 87, 167, 344,...

.-6, 5, -4, .9, .1, 16, 15, 47, 80, 177, 337,...

.11,-9, 13, -8, 15, -1, 32, 33, 97, 160, 351,...

-20,22,-21, 23,-16, 33, .1, 64, 63, 191, 320,...

On the main diagonal of this array we recognize the powers of 2, A131577.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,3,-2).

Formula

a(n) = 2^n/3 - 1/2 + (-1)^n/6 + A010892(n-1). - R. J. Mathar, Oct 06 2009

G.f.: x*(1 - x - 2*x^2 + 3*x^3)/((x-1) * (2*x-1) * (1+x) * (x^2-x+1)). - R. J. Mathar, Oct 06 2009

Mathematica

LinearRecurrence[{3, -2, -1, 3, -2}, {0, 1, 2, 2, 4}, 40] (* Harvey P. Dale, Mar 13 2014 *)
CoefficientList[Series[x (1 - x - 2 x^2 + 3 x^3)/((x - 1) (2 x - 1) (1 + x) (x^2 - x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

Prog

(PARI) concat(0, Vec(x*(1-x-2*x^2+3*x^3)/((x-1)*(2*x-1)*(1+x)*(x^2-x+1))+O(x^50))) \\ Jinyuan Wang, Feb 26 2020

Crossrefs

Sequence in context: A052935 A365091 A246789 * A052925 A006182 A267339

Adjacent sequences: A166019 A166020 A166021 * A166023 A166024 A166025

Keyword

nonn,easy

Author

Paul Curtz, Oct 04 2009

Extensions

Extended by R. J. Mathar, Oct 06 2009

Status

approved