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%I #22 Jun 06 2019 07:20:13
%S 8,3,0,1,9,8,5,1,7,0,6,7,8,2,3,9,3,4,5,5,2,2,5,6,2,7,1,9,5,5,2,7,1,0,
%T 6,5,7,7,8,2,0,6,3,0,8,4,3,9,4,5,4,3,7,3,1,9,3,9,5,5,2,4,1,2,2,1,6,0,
%U 8,4,8,3,2,0,4,5,6,1,8,8,9,6,2,2,6,4,1,6,3,8,6,9,7,2,6,2,9,1,2,1,5,9,1,2,3
%N Decimal expansion of the minimal zero x(1) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).
%C Let x(2) and x(3) denote the remaining zeros of F(x), x(2) < x(3). Then it could be proved that f(x(1)) = x(3), f(x(3)) = x(1), and f(x(2)) = x(2).
%C The decimal expansions of x(2) and x(3) in A206291 and A216863 respectively are presented.
%C We note that the plot of the restriction of F(x) to the interval [-2,2] "is very similar" to the plot of the polynomial (x-x(1))*(x-x(2))*(x-x(3)) for x in [-2,2].
%C Let A = {x in R: f^n(x) = x(2) for some nonnegative integer n, where f^n denotes the n-th iteration of f}. Then if z is a real number, which does not belong to A, and z(0):= z, z(n+1) = f(z(n)) = sqrt(2)*sin(Pi/4 - z(n)), n in N, then one of the subsequences either {z(2*n-1)} or {z(2*n)} is convergent to x(1) and the second one is convergent to x(3).
%H R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", <a href="https://www.jstor.org/stable/40391181">An Iteration Convergence: 11318[2007,745]</a>, Amer. Math. Monthly, 116 No 7 (2009), 648-649.
%e We have x(1) = -0.830198517...
%Y Cf. A216863, A206291, A215668, A215670, A215832, A215833, A168546.
%K nonn,cons
%O 0,1
%A _Roman Witula_, Sep 19 2012
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