A322171 Expansion of x*(3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

3, 11, 17, 19, 23, 31, 37, 39, 43, 51, 57, 59, 63, 71, 77, 79, 83, 91, 97, 99, 103, 111, 117, 119, 123, 131, 137, 139, 143, 151, 157, 159, 163, 171, 177, 179, 183, 191, 197, 199, 203, 211, 217, 219, 223, 231, 237, 239, 243, 251, 257, 259, 263, 271, 277, 279, 283, 291, 297, 299

Offset: 1

Author

Mark A. Thomas, Nov 29 2018

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A228826.

Programs

Magma

I:=[3,11,17,19]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Dec 06 2018

Maple

seq(coeff(series(x*(x^3+x^2+5*x+3)/((1-x)^2*(1+x^2)),x,n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 06 2018

Mathematica

CoefficientList[Series[(x^3 + x^2 + 5 x + 3)/((x - 1)^2 (x^2 + 1)), {x, 0, 50}], x] (* or *)
a[n_]:= (1/2) (10 n - (1 + 2 * I) (-I)^n - (1 - 2 I) I^n); Simplify[Array[a, 50]] (* Stefano Spezia, Nov 29 2018 *)
LinearRecurrence[{2, -2, 2, -1}, {3, 11, 17, 19}, 60] (* Vincenzo Librandi, Dec 06 2018 *)

PARI

Vec((3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)) + O(x^60)) \\ Andrew Howroyd, Nov 29 2018

Formula

a(n) = (1/2)*(10*n - (1+2*i)*(-i)^n - (1-2*i)*i^n), where i = sqrt(-1).

a(n) = 5*n - 2*sin(Pi*n/2) - cos(Pi*n/2).

a(n) = 5*n - A228826(n-1). - Andrew Howroyd, Nov 29 2018

G.f.: x*(x^3 + x^2 + 5*x + 3) / ((x - 1)^2 *(x^2 + 1)). - Vincenzo Librandi, Dec 06 2018