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A112712 Expansion of x/(1 - x + 2*x^2 - 2*x^3 + 2*x^4 - x^5 + x^6).

%I A112712 #18 Jun 14 2019 12:27:49
%S A112712 0,1,1,-1,-1,1,0,-2,0,2,0,-1,1,1,-1,-1,0,0,0,0,0,1,1,-1,-1,1,0,-2,0,2,
%T A112712 0,-1,1,1,-1,-1,0,0,0,0,0,1,1,-1,-1,1,0,-2,0,2,0,-1,1,1,-1,-1,0,0,0,0
%N A112712 Expansion of x/(1 - x + 2*x^2 - 2*x^3 + 2*x^4 - x^5 + x^6).
%C A112712 A modified Chebyshev transform of the Fibonacci numbers F(n) under the mapping g(x) -> (1/(1 + x^2)^2)*g(x/(1 + x^2)).
%H A112712 The Knot Atlas, <a href="http://katlas.math.toronto.edu/wiki/L6a2">L6a2</a>
%H A112712 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,-2,2,-2,1,-1).
%F A112712 G.f.: x*(1 + x - x^2 - x^3 + x^4 - 2*x^6 + 2*x^8 - x^10 + x^11 + x^12 - x^13 - x^14)/(1 - x^20).
%F A112712 a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + a(n-5) - a(n-6).
%F A112712 a(n) = a(n-20).
%F A112712 a(n) = Sum_{k=0..floor((n+2)/2)} (-1)^(k + 1)*C(n - k + 2, k - 1)*F(n-2k+2).
%F A112712 a(n) = Sum_{k=0..n} (F(k)*(-1)^((n - k)/2)*(Sum{j=0..n} C((j + k)/2, k)*(1 + (-1)^(n - j))(1 + (-1)^(j - k))/4)).
%F A112712 G.f.: -1/(x^(13/2)*f(x)), where f(x) = -1/x^(3/2) + 1/x^(5/2) - 2/x^(7/2) + 2/x^(9/2) - 2/x^(11/2) + 1/x^(13/2) - 1/x^(15/2) is the Jones polynomial for the link with Dowker-Thistlethwaite notation L6a2. - _Roger L. Bagula_, Jun 06 2007
%t A112712 f[x_] = -1/x^(3/2) + 1/x^(5/2) - 2/x^(7/2) + 2/x^(9/2) - 2/x^(11/2) + 1/x^(13/2) - 1/x^(15/2);
%t A112712 CoefficientList[Series[-(1/(x^(13/2)*f[x])), {x, 0, 50}], x] (* _Roger L. Bagula_, Jun 06 2007 *)
%t A112712 LinearRecurrence[{1,-2,2,-2,1,-1},{0,1,1,-1,-1,1},80] (* _Harvey P. Dale_, Jun 14 2019 *)
%o A112712 (Maxima) a(n) := sum((-1)^(k + 1)*binomial(n - k + 2, k - 1)*fib(n - 2*k + 2), k, 0, floor((n + 2)/2)); /* _Franck Maminirina Ramaharo_, Jan 08 2019 */
%Y A112712 Cf. A112713.
%K A112712 easy,sign
%O A112712 0,8
%A A112712 _Paul Barry_, Sep 15 2005