A098178 Expansion of (1+x)(1-x+x^2)/((1-x)(1+x^2)).

1, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2

Offset: 0

Author

Paul Barry, Aug 30 2004

Keywords

• easy
• nonn
• mult

Comments

Transform of A011782 under the Chebyshev mapping g(x)-> ((1-x^2)/(1+x^2)) * g(x/(1+x^2)).

Binomial transform is A098179.

Multiplicative with a(2) = 0, a(2^e) = 2 if e >= 2, a(p^e) = 1. [David W. Wilson, Jun 12 2005]

1, followed by period 4, repeat [1, 0, 1, 2]. [Joerg Arndt, Jan 06 2014]

Links

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

Crossrefs

Cf. A007877, A098179.

Programs

Magma

[1] cat &cat [[1, 0, 1, 2]^^30]; // Wesley Ivan Hurt, Jul 07 2016

Maple

with(numtheory); A098178:=n->signum(n)-1+sqrt((n-2)^2 mod 8); seq(A098178(n), n=0..100); # Wesley Ivan Hurt, Jan 04 2014

Mathematica

CoefficientList[Series[(1+x)(1-x+x^2)/((1-x)(1+x^2)),{x,0,120}],x] (* or *) PadRight[{1},120,{2,1,0,1}] (* Harvey P. Dale, May 01 2013 *)
Table[Sign[n] - 1 + Sqrt[Mod[(n - 2)^2, 8]], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 04 2014 *)
Join[{1},LinearRecurrence[{1, -1, 1},{1, 0, 1},104]] (* Ray Chandler, Sep 03 2015 *)

Formula

G.f.: (1+x)(1-x+x^2)/((1-x)(1+x^2)).

a(n) = 1 + cos(Pi*n/2) - 0^n.

a(n) = a(n-1) - a(n-2) + a(n-3) for n>2.

a(n) = A007877(n+2), n>0. Dirichlet g.f. (1-1/2^s+2/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011

a(n) = sign(n) - 1 + sqrt((n-2)^2 mod 8). - Wesley Ivan Hurt, Jan 04 2014

a(n) = a(n-4) for n>4. - Wesley Ivan Hurt, Jul 07 2016

E.g.f.: exp(x) + cos(x) - 1. - Ilya Gutkovskiy, Jul 07 2016