A180619 Consider the function f(n)=1/(Abs(n-r)), where r is the Dottie number, A003957. Let g(n) be defined by the recursion g(n)=Cos(g(n-1)),g(0)=1. Now, a(n)=floor(f(g(n)))

3, 5, 8, 11, 18, 26, 40, 58, 88, 130, 194, 287, 427, 633, 941, 1396, 2074, 3078, 4571, 6785, 10073, 14954, 22200, 32957, 48926, 72632, 107826, 160071, 237631, 352771, 523702, 777453, 1154157, 1713385, 2543579, 3776029, 5605645, 8321770, 12353952

Offset

0,1

Comments

This sequence gives a sense of the rate of convergence to the Dottie Number.

Because higher values of a(n) means that g(n) is converging to the Dottie number, quick convergence means a high rate of increase for a(n).

This can be compared to other methods for approximation the Dottie number, by defining an analogous sequence.

This gives us an algorithm to measure the rate of convergence, for ANY function that convergence to a constant.

a(n) is asymptotically approaches an exponential regression.

Links

Table of n, a(n) for n=0..38.

Example

For n=3, g(3)=cos(cos(cos(1)))
f(g(3))~=11.7931005 So a(3)=floor(11.7931005)=11.

Crossrefs

Sequence in context: A308266 A320594 A227563 * A196204 A220483 A136684

Adjacent sequences: A180616 A180617 A180618 * A180620 A180621 A180622

Keyword

nonn

Author

Ben Branman, Sep 12 2010

Status

approved