Feigenbaum Constant Approximations

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.

SEE ALSO: Almost Integer, Feigenbaum Constant