cp's OEIS Frontend

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

%I A106510 #36 Oct 28 2019 20:05:00
%S A106510 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 A106510 -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 A106510 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 A106510 Expansion of (1+x)^2/(1+x+x^2).
%C A106510 Row sums of the Riordan array ((1+x)/(1+x+x^2),x/(1+x)), A106509.
%C A106510 Equals INVERT transform of (1, -2, 3, -4, 5, ...). - _Gary W. Adamson_, Oct 10 2008
%H A106510 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1).
%F A106510 a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j)
%F A106510 a(n) = A049347(n-1) = A102283(n) if n >= 1. - _R. J. Mathar_, Aug 07 2011
%F A106510 From _Michael Somos_, Oct 15 2008: (Start)
%F A106510 Euler transform of length 3 sequence [ 1, -2, 1].
%F A106510 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 A106510 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 A106510 a(-n) = a(n). a(n+3) = a(n) unless n = 0 or n = -3.
%F A106510 a(n) = Sum_{k=0..n} A128908(n,k)*(-1)^(n-k). - _Philippe Deléham_, Jan 22 2012
%e A106510 1 + x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + x^13 - x^14 + ...
%t A106510 {1}~Join~LinearRecurrence[{-1, -1}, {1, -1}, 105] (* _Jean-François Alcover_, Oct 28 2019 *)
%o A106510 (PARI) {a(n) = if( n==0, 1, [0, 1, -1][n%3 + 1])} \\ _Michael Somos_, Oct 15 2008
%o A106510 (PARI) {a(n) = if( n==0, 1, kronecker(-3, n))} \\ _Michael Somos_, Oct 15 2008
%o A106510 (PARI) A106510(n)=kronecker(-3, n+!n) \\ _M. F. Hasler_, May 07 2018
%K A106510 easy,sign,mult
%O A106510 0,1
%A A106510 _Paul Barry_, May 04 2005
%E A106510 Edited by _M. F. Hasler_, May 07 2018