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

0,2

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).

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

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 *)

Crossrefs

Cf. A099486, A099487, A099488.

Sequence in context: A096147 A225431 A359248 * A077830 A106460 A319512

Adjacent sequences: A099486 A099487 A099488 * A099490 A099491 A099492

Keyword

easy,nonn

Author

Paul Barry, Oct 18 2004

Status

approved