cp's OEIS Frontend

For n=0, the only possible solution is f(0)=0, which yields a(0)=1.

f(n)->1 as n->infinity.

a(n) ~ -n/log(cos(1))

f(1) = the Dottie number 0.73908513321516 = A003957

f(2) is A125578

f(3) is A125579

a(n) is defined for negative values of n as well.

If we let a(n)=floor[c(n)], c(n)=1/(1-f(n)), then f(n)^n=cos(f(n)) <=> 1-1/c(n) = cos(1-1/c(n))^(1/n) = exp(log(cos(1-1/c(n)))/n) = exp(log(cos(1)+O(1/c(n)^2))/n) = 1+log(cos(1))/n+o(1/n), assuming c(n) ~ c*n, which then yields c = -1/log(cos(1)). - M. F. Hasler, Mar 05 2012

The fixed point solution for the composite function y = cos(sin(x)).

The value A131691 is equal to the arccosine of this value and this value is equal to the arcsine of A131691.

This is analogous to A003957, the fixed point of the cosine function in radians and A330119, the fixed point of the cosine function in degrees. Each of the three are the unique real solutions to cos(x)-x=0, in their respective angular units. The quadrant unit offers a nice symmetry, cos(0)=1 and cos(1)=0. Unlike the previous two, the quadrant fixed point is not an attractor of its cosine function. It cannot be found by iterative cosine application. Although not proven, iterative quadrant cosine can be seen empirically to diverge for all initial values.

A graphical solution can be demonstrated by plotting y = cos(x*Pi/2) - x, which shows a single zero near x=0.6.

The bisection method converges for the entire range of the cosine function (-1 to 1). Newton's method also converges with reasonable initial estimate.