A216863 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).

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, 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, 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

Offset: 1

Author

Roman Witula, Sep 18 2012

Keywords

• nonn
• cons

Comments

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.

Examples

			We have x(3) = 1.4127945857276222... < sqrt(2).
		

Links

• R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007, 745], Amer. Math. Monthly, 116 No 7 (2009), 648-649.

Crossrefs

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

Programs

Mathematica

f[x_] := Cos[x] - Sin[x];
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 *)

PARI

default(realprecision,110);
f(x) = cos(x) - sin(x);
solve(x = 1.4, 1.5, f(f(x)) - x) \\ Rick L. Shepherd, Jan 03 2014

Extensions

Offset corrected by Rick L. Shepherd, Jan 03 2014