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A213043 Convolution of (1,-1,2,-2,3,-3,...) and A000045 (Fibonacci numbers).

%I A213043 #18 Mar 16 2016 12:52:15
%S A213043 1,0,3,1,7,5,16,17,38,50,94,138,239,370,617,979,1605,2575,4190,6755,
%T A213043 10956,17700,28668,46356,75037,121380,196431,317797,514243,832025,
%U A213043 1346284,2178293,3524594,5702870,9227482,14930334,24157835,39088150,63246005,102334135
%N A213043 Convolution of (1,-1,2,-2,3,-3,...) and A000045 (Fibonacci numbers).
%C A213043 (1,-1,2,-2,3,-3,...) = ((-1)^n)*(1+floor(n/2)), which results from A001057 by removing its initial 0.
%H A213043 Colin Barker, <a href="/A213043/b213043.txt">Table of n, a(n) for n = 0..1000</a>
%H A213043 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1,-2,-1).
%F A213043 a(n) = 3*a(n-2)+a(n-3)-2*a(n-4)-a(n-5).
%F A213043 G.f.: 1/((1 + x)^2 * (1 - 2*x + x^3)).
%F A213043 From _Vladimir Reshetnikov_, Oct 29 2015: (Start)
%F A213043 a(n) = Fibonacci(n+1) + ((-1)^n*(2*n+1)-1)/4, where Fibonacci(n) = A000045(n).
%F A213043 Recurrence (4-term): a(0) = 1, a(1) = 0, a(2) = 3, (2*n+1)*a(n) = n + 1 - 2*a(n-1) + 4*(n+1)*a(n-2) + (2*n+3)*a(n-3).
%F A213043 (End)
%F A213043 From _Colin Barker_, Mar 16 2016: (Start)
%F A213043 a(n) = (-5-5*(-1)^n+2^(1-n)*sqrt(5)*(-(1-sqrt(5))^(1+n)+(1+sqrt(5))^(1+n))+10*(-1)^n*(1+n))/20.
%F A213043 a(n) = (sqrt(5)*2^(1-n)*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))+10*(n+1)-10)/20 for n even.
%F A213043 a(n) = (sqrt(5)*2^(1-n)*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))-10*(n+1))/20 for n odd.
%F A213043 (End)
%e A213043 a(5) = (1,-1,2,-2,3,-3)**(1,1,2,3,5,8)=1*8-1*5+2*3-2*2+3*1-3*1 = 5.
%t A213043 f[x_] := (1 - x^2) (1 + x); g[x] := 1 - x - x^2;
%t A213043 s = Normal[Series[1/(f[x] g[x]), {x, 0, 60}]]
%t A213043 c = CoefficientList[s, x]  (* A213043 *)
%t A213043 LinearRecurrence[{0, 3, 1, -2, -1}, {1, 0, 3, 1, 7}, 60]
%t A213043 Table[Fibonacci[n+1] + ((-1)^n (2n+1) - 1)/4, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 29 2015 *)
%o A213043 (PARI) Vec(1/((1-x)*(1+x)^2*(1-x-x^2)) + O(x^50)) \\ _Colin Barker_, Mar 16 2016
%Y A213043 Cf. A000045, A213500.
%K A213043 nonn,easy
%O A213043 0,3
%A A213043 _Clark Kimberling_, Jun 10 2012