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%I #56 May 20 2023 23:16:27
%S 0,1,-1,1,-1,0,-1,-1,17,587,3151,-173,-2641109,-6343201,29002301,
%T 24753572807,6013935944287,-979056822493,-11395219462649,
%U -4313800586682649,-2178360615103441,74893762899375939059,5307412498351127900521
%N Numerators of series coefficients for Archimedes's spiral that transforms into Galileo's spiral.
%C The curve defined by the differential equation in polar coordinates r'(t)^2 + r(t)^2 = t^2 with r(0)=0, r"(0) > 0. Solution is represented by a power series in z=t^2 (satisfying the differential equation 4*z*r'(z)^2 + r(z)^2 = z). The sequence lists coefficients of t^(2*n) (or z^n) in this series.
%C For large t, the curve represents Archimedes's spiral. As t vanishes, the curve transforms into a Galileo spiral. The junction point of the curve and the ray uniformly rotated in the origin coordinates moves uniformly accelerated.
%C Let L_{A) and L_{AG} are the lengths of Archimedean spiral and the spiral defined by the differential equation, then lim_{t -> oo} L_{A}/L_{AG} = 1. In other words, the lengths of Archimedean spiral and the spiral defined by the differential equation are equivalent for large t. - _Mikhail Gaichenkov_, Jan 08 2013
%C According to Robert Bryant, the key to understanding the solutions of the ODE near the singular points is the Briot-Bouquet normal form for dealing with singular points, and, fortunately, it is just the right thing both at the origin and along the lines theta^2 - r^2 = 0. - _Mikhail Gaichenkov_, Feb 18 2013
%H Robert Bryant, <a href="http://mathoverflow.net/questions/121402/">MathOverflow: What is symmetry group of non-linear equation?</a>
%H A. Pichugin, <a href="http://mathoverflow.net/questions/54393/analytical-solutions-of-a-differential-equation-from-archimedes-spiral/79737#79737">MathOverflow: Analytical solutions of a differential equation (from Archimedes' Spiral)</a>
%e The first ten terms of this expansion are: r(t) = 0 + 1/2*t^2 - 1/32*t^4 + 1/768*t^6 - 1/49152*t^8 + 0*t^10 - 1/56623104*t^12 - 1/317893824*t^14 + 17/541165879296*t^16 + 587/175337744891904*t^18 + ...
%e The radius of the convergence is about 7/2.
%p Order:=60: dsolve( { diff(r(t),t)^2 + r(t)^2 = t^2, r(0)=0 }, r(t), series ); # _Max Alekseyev_, Dec 19 2012
%t km = 23; a[0] = 0; r[t_] = Sum[a[k] t^(2 k), {k, 0, km}]; coes = CoefficientList[Series[r'[t]^2 + r[t]^2 - t^2 , {t, 0, 2 km}], t] // Union // Rest; Table[a[k], {k, 0, km}] /. Solve[Thread[coes == 0] ] // Last // Most // Numerator (* _Jean-François Alcover_, Jan 18 2013 *)
%Y Denominators are listed in A202408.
%K sign,frac
%O 0,9
%A _Mikhail Gaichenkov_, Dec 19 2011
%E Corrected and extended by _Max Alekseyev_, Dec 19 2011
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