Feigenbaum Constant Approximations | |

A curious approximation to the Feigenbaum constant is given by

(1) |

where is Gelfond's constant, which is good to 6 digits to the right of the decimal point.

M. Trott (pers. comm., May 6, 2008) noted

(2) |

where is Gauss's constant, which is good to 4 decimal digits, and

(3) |

where is the tetranacci constant, which is good to 3 decimal digits.

A strange approximation good to five digits is given by the solution to

(4) |

which is

(5) |

where is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).

(6) |

gives to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).

M. Hudson (pers. comm., Nov. 20, 2004) gave

(7) | |||

(8) | |||

(9) |

which are good to 17, 13, and 9 digits respectively.

Stoschek gave the strange approximation

(10) |

which is good to 9 digits.

R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations

(11) | |||

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) |

where *e* is the base of the natural logarithm and is Gelfond's constant, which are good to 3, 3, 5, 7, 9, and 10 decimal digits, respectively, and

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) | |||

(22) | |||

(23) |

which are good to 3, 3, 3, 4, 6, 8, and 8 decimal digits, respectively.

An approximation to due to R. Phillips (pers. comm., Jan. 27, 2005) is obtained by numerically solving

(24) |

for , where is the golden ratio, which is good to 4 digits.