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

Author

Paul Barry, Sep 07 2009

Keywords

• easy
• sign

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).

Crossrefs

Cf. A100050 (first differences).

Hankel transform of A165201.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x)/(1-x+x^2)^2 )); // G. C. Greubel, Jul 18 2019

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 *)

PARI

my(x='x+O('x^70)); Vec((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

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