A168056 Expansion of (1+2*x^2+x^3)/((1-x)^2*(1+x+x^2)).

1, 1, 3, 5, 5, 7, 9, 9, 11, 13, 13, 15, 17, 17, 19, 21, 21, 23, 25, 25, 27, 29, 29, 31, 33, 33, 35, 37, 37, 39, 41, 41, 43, 45, 45, 47, 49, 49, 51, 53, 53, 55, 57, 57, 59, 61, 61, 63, 65, 65, 67, 69, 69, 71, 73, 73, 75, 77, 77, 79, 81, 81, 83, 85, 85, 87, 89, 89, 91, 93, 93

Offset: 0

Author

Paul Barry, Nov 17 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A168053.

Programs

Magma

I:=[1,1,3,5]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Jul 08 2016

Mathematica

LinearRecurrence[{1, 0, 1, -1}, {1, 1, 3, 5}, 100] (* G. C. Greubel, Jul 07 2016 *)
CoefficientList[Series[(1 + 2 x^2 + x^3) / ((1 - x)^2 (1 + x + x^2)), {x, 0, 80}], x] (* Vincenzo Librandi, Jul 08 2016 *)

Formula

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

a(n) = A168057(n)/2^n.

a(n) = (12*n+3+6*cos(2*n*Pi/3)-2*sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017