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A106510 Expansion of (1+x)^2/(1+x+x^2). 12

%I

%S 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,

%T -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,

%U 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

%N Expansion of (1+x)^2/(1+x+x^2).

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

%C Equals INVERT transform of (1, -2, 3, -4, 5, ...). - _Gary W. Adamson_, Oct 10 2008

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1).

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

%F a(n) = A049347(n-1) = A102283(n) if n >= 1. - _R. J. Mathar_, Aug 07 2011

%F From _Michael Somos_, Oct 15 2008: (Start)

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

%F 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).

%F 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)

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

%F a(n) = Sum_{k=0..n} A128908(n,k)*(-1)^(n-k). - _Philippe Deléham_, Jan 22 2012

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

%o (PARI) {a(n) = if( n==0, 1, [0, 1, -1][n%3 + 1])} \\ _Michael Somos_, Oct 15 2008

%o (PARI) {a(n) = if( n==0, 1, kronecker(-3, n))} \\ _Michael Somos_, Oct 15 2008

%o (PARI) A106510(n)=kronecker(-3, n+!n) \\ _M. F. Hasler_, May 07 2018

%K easy,sign,mult

%O 0,1

%A _Paul Barry_, May 04 2005

%E Edited by _M. F. Hasler_, May 07 2018

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Last modified October 13 16:50 EDT 2019. Contains 327968 sequences. (Running on oeis4.)