A272071 Expansion of x*(3 - 2*x + x^2)/((1 - x)^2*(1 + x + x^2)).

0, 3, 1, 2, 5, 3, 4, 7, 5, 6, 9, 7, 8, 11, 9, 10, 13, 11, 12, 15, 13, 14, 17, 15, 16, 19, 17, 18, 21, 19, 20, 23, 21, 22, 25, 23, 24, 27, 25, 26, 29, 27, 28, 31, 29, 30, 33, 31, 32, 35, 33, 34, 37, 35, 36, 39, 37, 38, 41, 39, 40, 43, 41, 42, 45, 43, 44, 47, 45, 46, 49, 47, 48, 51, 49, 50, 53, 51, 52, 55

Offset

0,2

Links

Table of n, a(n) for n=0..79.

Ilya Gutkovskiy, Illustration

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

O.g.f.: x*(3 - 2*x + x^2)/((1 - x)^2*(1 + x + x^2)).

E.g.f.: 2*(3*x*exp(x) + 3*exp(x) - exp(-x/2)*(3*cos((sqrt(3)*x)/2) - 4*sqrt(3)*sin((sqrt(3)*x)/2)))/9.

a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.

a(n) = 2*(3*n + 4*sqrt(3)*sin((2*Pi*n)/3) - 3*cos((2*Pi*n)/3) + 3)/9.

a(n) = 3*n - 2*floor(n/3) - 5*floor((n + 1)/3). - Vaclav Kotesovec, Apr 22 2016

a(n) mod 2 = A011655(n).

Maple

A272071:=proc(n) option remember;
if n=0 then 0 elif n=1 then 3 elif n=2 then 1 elif n=3 then 2 else
a(n-1)+a(n-3)-a(n-4); fi; end: seq(A272071(n), n=0..150); # Wesley Ivan Hurt, Apr 20 2016

Mathematica

LinearRecurrence[{1, 0, 1, -1}, {0, 3, 1, 2}, 80]
Table[2 ((3 n + 4 Sqrt[3] Sin[(2 Pi n)/3] - 3 Cos[(2 Pi n)/3] + 3)/9), {n, 0, 79}]
CoefficientList[Series[x*(3 - 2*x + x^2)/((1 - x)^2*(1 + x + x^2)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Apr 20 2016 *)

Prog

(PARI) concat(0, Vec(x*(3-2*x+x^2)/((1-x)^2*(1+x+x^2)) + O(x^99))) \\ Altug Alkan, Apr 22 2016

Crossrefs

Cf. A011655, A028242.

Sequence in context: A033765 A033777 A329144 * A033801 A332319 A359753

Adjacent sequences: A272068 A272069 A272070 * A272072 A272073 A272074

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Apr 19 2016

Status

approved