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A145064 Reduced numerators of the first convergent to the cube root of n using the recursion x = (2*x+n/x^2)/3.

%I A145064 #16 Oct 28 2017 10:47:07
%S A145064 2,1,4,5,2,7,8,3,10,11,4,13,14,5,16,17,6,19,20,7,22,23,8,25,26,9,28,
%T A145064 29,10,31,32,11,34,35,12,37,38,13,40,41,14,43,44,15,46,47,16,49,50,17,
%U A145064 52,53,18,55,56,19,58,59,20,61,62,21,64,65,22,67,68,23,70,71,24,73,74,25
%N A145064 Reduced numerators of the first convergent to the cube root of n using the recursion x = (2*x+n/x^2)/3.
%C A145064 The same as A051176 without the first two terms.
%H A145064 Colin Barker, <a href="/A145064/b145064.txt">Table of n, a(n) for n = 0..1000</a>
%H A145064 Cino Hilliard, <a href="http://groups.google.com/group/roots-by-recursion/web/recursion-on-polynomial">Roots by Recursion</a> [broken link]
%H A145064 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A145064 The recursion was derived experimentally by analyzing the patterns of root recursions for polynomials
%F A145064 f(x) = a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) and
%F A145064 g(x) = a(n-1)x^(n-1)+a(n-2)x^(n-2)+...+a(2)x+a(1)
%F A145064 where the recursion x = a(0)/g(x) may or may not converge to a root and many iterations are required to get greater accuracy. By introducing an averaging scheme, a root is found if it exists and convergence is much faster to a root of f(x) See the link for details. This cubic recursion is equivalent to Newton's Method.
%F A145064 From _Colin Barker_, Feb 02 2016: (Start)
%F A145064 a(n) = 2*a(n-3)-a(n-6) for n>5.
%F A145064 G.f.: (2+x+4*x^2+x^3-x^5) / ((1-x)^2*(1+x+x^2)^2). (End)
%t A145064 CoefficientList[Series[(2+x+4*x^2+x^3-x^5)/((1-x)^2*(1+x+x^2)^2), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 26 2017 *)
%o A145064 (PARI)
%o A145064   rroot3(d,p) = /* Find a root of x^3 - d */Q {
%o A145064   local(x=1,x1=1,j);
%o A145064   for(j=1,p,
%o A145064   x=(x1+x+d/x^2)/3; /* average scheme for a cube root of d */
%o A145064   x1=x; print1(numerator(x)",");
%o A145064   );
%o A145064 }
%o A145064 for(k=0,100,rroot3(k,1))
%o A145064 (PARI) Vec((2+x+4*x^2+x^3-x^5)/((1-x)^2*(1+x+x^2)^2) + O(x^100)) \\ _Colin Barker_, Feb 02 2016
%Y A145064 Cf. A051176
%K A145064 frac,nonn,easy
%O A145064 0,1
%A A145064 _Cino Hilliard_, Sep 30 2008