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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 (list; graph; refs; listen; history; text; internal format)
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 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

LINKS

Table of n, a(n) for n=0..22.

Robert Bryant, MathOverflow: What is symmetry group of non-linear 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 (* Jean-Fran├žois Alcover, Jan 18 2013 *)

CROSSREFS

Denominators are listed in A202408.

Sequence in context: A056771 A041547 A041544 * A009709 A094133 A162490

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

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Last modified September 18 17:06 EDT 2014. Contains 246924 sequences.