A099489 Expansion of (1-x+x^2)/((1+x^2)(1-4x+x^2)).

1, 3, 11, 42, 157, 585, 2183, 8148, 30409, 113487, 423539, 1580670, 5899141, 22015893, 82164431, 306641832, 1144402897, 4270969755, 15939476123, 59486934738, 222008262829, 828546116577, 3092176203479, 11540158697340

Offset: 0

Author

Paul Barry, Oct 18 2004

Keywords

• easy
• nonn

Comments

A Chebyshev transform of the sequence A002001 which has with g.f. (1-x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

• Matthew House, Table of n, a(n) for n = 0..1739
• Index entries for linear recurrences with constant coefficients, signature (4,-2,4,-1).

Crossrefs

Cf. A099486, A099487, A099488.

Programs

Mathematica

CoefficientList[Series[(1-x+x^2)/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *_)
LinearRecurrence[{4,-2,4,-1},{1,3,11,42},30] (* Harvey P. Dale, Dec 28 2019 *)

Formula

a(n) = 4*a(n-1)-2*a(n-2)+4*a(n-3)-a(n-4). - corrected by Matthew House, Oct 22 2016

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(3*4^(n-2*k)+0^(n-2*k))/4.

a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3)).

a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*A001353(n-k+1).

a(n) = 3*A001353(n+1)/4 +A056594(n)/4. - R. J. Mathar, Sep 21 2012