A216863 - OEIS

%I #40 Jan 15 2014 04:46:00

%S 1,4,1,2,7,9,4,5,8,5,7,2,7,6,2,2,2,4,5,2,9,3,5,7,7,5,8,6,5,0,6,6,3,7,

%T 6,3,2,1,2,4,5,5,8,8,9,1,2,7,3,8,1,5,1,6,5,8,6,4,1,7,7,5,2,5,5,3,0,1,

%U 1,2,5,2,4,7,7,1,2,2,0,5,5,1,8,2,4,3,3,4,0,8,7,8,5,0,0,6,0,8,9,2,5,6,1,5,0

%N The decimal expansion of the maximal zero x(3) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

%C The decimal expansions of the only other zeros x(1) and x(2) of F(x) are given in A216891 and A206291. See the comments in A216891 for more information about intriguing properties of these zeros.

%H R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", <a href="http://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(3) = 1.4127945857276222... < sqrt(2).

%t f[x_] := Cos[x] - Sin[x];

%t FindRoot[f[f[x]] - x, {x, 1.4}, WorkingPrecision -> 110] (* _T. D. Noe_, Sep 24 2012; first line by _Rick L. Shepherd_, Jan 04 2014 *)

%o (PARI)

%o default(realprecision,110);

%o f(x) = cos(x) - sin(x);

%o solve(x = 1.4, 1.5, f(f(x)) - x) \\ _Rick L. Shepherd_, Jan 03 2014

%Y Cf. A216891, A206291, A215668, A215670, A215832, A215833, A168546.

%K nonn,cons

%O 1,2

%A _Roman Witula_, Sep 18 2012

%E Offset corrected by _Rick L. Shepherd_, Jan 03 2014