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

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

Offset: 1

Author

Eric W. Weisstein, Dec 22 2003

Keywords

• sign
• easy

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)

Examples

			t + t^3/6 - t^5/40 + t^7/112 - (5*t^9)/1152 + (7*t^11)/2816 - ...
		

Links

• Eric Weisstein's World of Mathematics, Archimedes' Spiral

Crossrefs

Denominators are in A002595.