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A117569 Expansion of (1+x+x^2)/(1+x^2).

%I A117569 #32 Dec 12 2016 18:22:36
%S A117569 1,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,
%T A117569 -1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,
%U A117569 1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0
%N A117569 Expansion of (1+x+x^2)/(1+x^2).
%C A117569 Row sums of A117568.
%H A117569 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A117569 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1).
%F A117569 G.f.: (1-x^3)/((1-x)(1+x^2)); a(n)=0^n+(1-(-1)^n)(cos(pi*n/2)+sin(pi*n/2))/2;
%F A117569 a(n) = A101455(n), n>0. - _R. J. Mathar_, Aug 10 2008
%F A117569 Expansion of (1 - x^2) * (1 - x^3) / ((1 - x) * (1 - x^4)) in powers of x.
%F A117569 G.f.: 1 / (1 - x / (1 + x / (1 - x / (1 + x)))). - _Michael Somos_, Apr 02 2012
%F A117569 Euler transform of length 4 sequence [ 1, -1, -1, 1]. - _Michael Somos_, Aug 04 2009
%F A117569 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v - u * (2 - u) * (2*v - 1).
%F A117569 a(n) is completely multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4).
%F A117569 a(2*n) = 0 unless n=0, a(4*n + 3) = -1, a(4*n + 1) = a(0) = 1.
%F A117569 a(-n) = -a(n) unless n=0. a(n+2) = -a(n) unless n=0 or n=-2.
%F A117569 a(n) = -A163805(n) unless n=0. a(n) = (-1)^n * A163805(n). Convolution inverse of A163804.
%F A117569 E.g.f.: 1 + sin(x). - _Arkadiusz Wesolowski_, Aug 13 2012
%F A117569 a(n) = floor(1/(n+1)) + (1-(-1)^n)/2*(-1)^((n-1)/2). - _Tani Akinari_, Nov 09 2012
%e A117569 1 + x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + ...
%t A117569 CoefficientList[Series[(1+x+x^2)/(1+x^2),{x,0,120}],x] (* or *) LinearRecurrence[{0,-1},{1,1,0},120] (* or *) PadRight[{1},120,{0,1,0,-1}] (* _Harvey P. Dale_, Dec 12 2016 *)
%o A117569 (PARI) a(n) = (n==0) + [0, 1, 0, -1][n%4 + 1] /* _Michael Somos_, Aug 04 2009 */
%o A117569 (PARI) a(n) = (n==0) + kronecker( -4, n) /* _Michael Somos_, Aug 04 2009 */
%Y A117569 Cf. A073097, A163804, A163805.
%K A117569 easy,sign,mult
%O A117569 0,1
%A A117569 _Paul Barry_, Mar 29 2006