login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 7 07:15 EST 2016. Contains 278841 sequences.