%I
%S 1,1,1,1,5,7,21,11,429,715,2431,4199,29393,52003,185725,334305,
%T 3231615,3535767,64822395,39803225,883631595,1641030105,407771117,
%U 11435320455,171529806825,107492012277,1215486600363,2295919134019
%N Numerator of Maclaurin expansion of (t*sqrt(t^2+1) + arcsinh(t))/2, the arc length of Archimedes' spiral.
%C From _Mikhail Gaichenkov_, Feb 05 2013: (Start)
%C For Archimedean spiral (r=at) and the arc length s(t)= a(t*sqrt(t^2+1) + arcsinh(t))/2, the limit of s’’(t)=a, t > infinity. In other words, a point moves with uniform acceleration along the spiral while the spiral corresponds to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity.
%C The error of approximation for large t: as’’(t) ~ a/(2(1+t^2)) (Gaichenkov private research).
%C The arc of the Archimedean spiral is approximated by the differential equation in polar coordinates r’^2+r^2=(at)^2 (see A202407). (End)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ArchimedesSpiral.html">Archimedes' Spiral</a>
%e t + t^3/6  t^5/40 + t^7/112  (5*t^9)/1152 + (7*t^11)/2816  ...
%Y Denominators are in A002595.
%K sign,easy
%O 1,5
%A _Eric W. Weisstein_, Dec 22 2003
