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.
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