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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
 1, 1, -1, 1, -5, 7, -21, 11, -429, 715, -2431, 4199, -29393, 52003, -185725, 334305, -3231615, 3535767, -64822395, 39803225, -883631595, 1641030105, -407771117, 11435320455, -171529806825, 107492012277, -1215486600363, 2295919134019 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS From Mikhail Gaichenkov, Feb 05 2013: (Start) 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. The error of approximation for large t: |a-s’’(t)| ~ a/(2(1+t^2)) (Gaichenkov private research). The arc of the Archimedean spiral is approximated by the differential equation in polar coordinates r’^2+r^2=(at)^2 (see A202407). (End) LINKS Eric Weisstein's World of Mathematics, Archimedes' Spiral EXAMPLE t + t^3/6 - t^5/40 + t^7/112 - (5*t^9)/1152 + (7*t^11)/2816 - ... CROSSREFS Denominators are in A002595. Sequence in context: A192422 A120035 A198302 * A057424 A027152 A076197 Adjacent sequences:  A091151 A091152 A091153 * A091155 A091156 A091157 KEYWORD sign,easy AUTHOR Eric W. Weisstein, Dec 22 2003 STATUS approved

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Last modified June 20 09:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)