A106510 Expansion of (1+x)^2/(1+x+x^2).

1, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1

Offset

0,1

Comments

Row sums of the Riordan array ((1+x)/(1+x+x^2),x/(1+x)), A106509.

Equals INVERT transform of (1, -2, 3, -4, 5, ...). - Gary W. Adamson, Oct 10 2008

Links

Table of n, a(n) for n=0..104.

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

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j)

a(n) = A049347(n-1) = A102283(n) if n >= 1. - R. J. Mathar, Aug 07 2011

From Michael Somos, Oct 15 2008: (Start)

Euler transform of length 3 sequence [ 1, -2, 1].

a(n) is multiplicative with a(3^e) = 0^e, a(p^e) = 1 if p == 1 (mod 3), a(p^e) = (-1)^e if p == 2 (mod 3).

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v) = 4 - 3*v - u * (4 - 2*v - u). (End)

a(-n) = a(n). a(n+3) = a(n) unless n = 0 or n = -3.

a(n) = Sum_{k=0..n} A128908(n,k)*(-1)^(n-k). - Philippe Deléham, Jan 22 2012

Example

1 + x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + x^13 - x^14 + ...

Mathematica

{1}~Join~LinearRecurrence[{-1, -1}, {1, -1}, 105] (* Jean-François Alcover, Oct 28 2019 *)

Prog

(PARI) {a(n) = if( n==0, 1, [0, 1, -1][n%3 + 1])} \\ Michael Somos, Oct 15 2008
(PARI) {a(n) = if( n==0, 1, kronecker(-3, n))} \\ Michael Somos, Oct 15 2008
(PARI) A106510(n)=kronecker(-3, n+!n) \\ M. F. Hasler, May 07 2018

Crossrefs

Sequence in context: A156707 A131309 A267208 * A163806 A163810 A163804

Adjacent sequences: A106507 A106508 A106509 * A106511 A106512 A106513

Keyword

easy,sign,mult

Author

Paul Barry, May 04 2005

Extensions

Edited by M. F. Hasler, May 07 2018

Status

approved