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

1,2

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.

Links

Table of n, a(n) for n=1..105.

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A241185 A243584 A084460 * A120578 A096249 A081454

Adjacent sequences: A216860 A216861 A216862 * A216864 A216865 A216866

Keyword

nonn,cons

Author

Roman Witula, Sep 18 2012

Extensions

Offset corrected by Rick L. Shepherd, Jan 03 2014

Status

approved