A165202 Expansion of (1+x)/(1 - x + x^2)^2.

1, 3, 3, -1, -6, -6, 1, 9, 9, -1, -12, -12, 1, 15, 15, -1, -18, -18, 1, 21, 21, -1, -24, -24, 1, 27, 27, -1, -30, -30, 1, 33, 33, -1, -36, -36, 1, 39, 39, -1, -42, -42, 1, 45, 45, -1, -48, -48, 1, 51, 51, -1, -54, -54, 1, 57, 57, -1, -60, -60, 1

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Formula

a(n) = cos(Pi*n/3) + sin(Pi*n/3)*(2n/3 + 1)*sqrt(3).

a(n) = A099254(n) + A099254(n-1). - R. J. Mathar, May 02 2013

Mathematica

LinearRecurrence[{2, -3, 2, -1}, {1, 3, 3, -1}, 70] (* G. C. Greubel, Jul 18 2019 *)
(-1)^Quotient[#-1, 3]{1, 1+#, #}[[Mod[#, 3, 1]]]&/@Range[0, 10] (* Federico Provvedi, Jul 18 2021 *)

Prog

(PARI) my(x='x+O('x^70)); Vec((1+x)/(1-x+x^2)^2) \\ G. C. Greubel, Jul 18 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x)/(1-x+x^2)^2 )); // G. C. Greubel, Jul 18 2019
(Sage) ((1+x)/(1-x+x^2)^2).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019
(GAP) a:=[1, 3, 3, -1];; for n in [5..70] do a[n]:=2*a[n-1]-3*a[n-2]+ 2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 18 2019

Crossrefs

Cf. A100050 (first differences).

Hankel transform of A165201.

Sequence in context: A174128 A131070 A295290 * A010468 A082009 A110640

Adjacent sequences: A165199 A165200 A165201 * A165203 A165204 A165205

Keyword

easy,sign

Author

Paul Barry, Sep 07 2009

Status

approved