

A202407


Numerators of series coefficients for Archimedes' spiral that transforms into Galileo's spiral.


2



0, 1, 1, 1, 1, 0, 1, 1, 17, 587, 3151, 173, 2641109, 6343201, 29002301, 24753572807, 6013935944287, 979056822493, 11395219462649, 4313800586682649, 2178360615103441, 74893762899375939059, 5307412498351127900521
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OFFSET

0,9


COMMENTS

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.
For large t, the curve represents Archimedes' 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.
Let L_{A) and L_{AG} are the lengths of Archimedean spiral and the spiral defined by the differential equation, then the limit lim_{t > infinity} L_{A}/L_{AG} = 1. In other words, the lengths of Atchimedean spiral and the spiral are equivalent for large t.  Mikhail Gaichenkov, Jan 08 2013
According to Robert Bryant, the key to understanding the solutions of the ODE near the singular points is the BriotBouquet 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


LINKS

Table of n, a(n) for n=0..22.
Robert Bryant, MathOverflow: What is symmetry group of nonlinear equation?
A. Pichugin, MathOverflow: Analytical solutions of a differential equation (from Archimedes' Spiral)


EXAMPLE

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 + ...
The radius of the convergence is about 7/2.


MAPLE

Order:=60: dsolve( { diff(r(t), t)^2 + r(t)^2 = t^2, r(0)=0 }, r(t), series ); # From Max Alekseyev, Dec 19 2012


MATHEMATICA

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 (* JeanFrançois Alcover, Jan 18 2013 *)


CROSSREFS

Denominators are listed in A202408.
Sequence in context: A056771 A041547 A041544 * A009709 A162490 A142744
Adjacent sequences: A202404 A202405 A202406 * A202408 A202409 A202410


KEYWORD

sign,frac


AUTHOR

Mikhail Gaichenkov, Dec 19 2011


EXTENSIONS

Corrected and extended by Max Alekseyev, Dec 19 2011


STATUS

approved



