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 A091154 Numerator of Maclaurin expansion of (t*sqrt(t^2+1) + arcsinh(t))/2, the arc length of Archimedes' spiral. 0

%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: |a-s’’(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

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Last modified July 23 22:49 EDT 2019. Contains 325278 sequences. (Running on oeis4.)