

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
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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: as’’(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

Table of n, a(n) for n=1..28.
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



