cp's OEIS Frontend

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'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.

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

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