A curious approximation to the Feigenbaum constant 
is given by
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(1)
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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
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(2)
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where 
is Gauss's constant, which is good to 4 decimal
digits, and
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(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
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(4)
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which is
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(5)
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where 
is the Lambert W-function (G. Deppe, pers.
comm., Feb. 27, 2003).
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(6)
|
gives 
to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).
M. Hudson (pers. comm., Nov. 20, 2004) gave
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(7)
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(8)
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(9)
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which are good to 17, 13, and 9 digits respectively.
Stoschek gave the strange approximation
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(10)
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which is good to 9 digits.
R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations
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(11)
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(12)
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(13)
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(14)
|
 |  | ![pi-tan^(-1)[(e-1)^(-16)-e^pi],](/images/equations/FeigenbaumConstantApproximations/Inline30.svg) |
(15)
|
 |  | ![(e(e-1))/(1+exp{8[(1+e^(-8))^(3/2)-2]})](/images/equations/FeigenbaumConstantApproximations/Inline33.svg) |
(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
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(17)
|
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(18)
|
 |  | ![tan[e-tan^(-1)(e^pi)]](/images/equations/FeigenbaumConstantApproximations/Inline43.svg) |
(19)
|
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(20)
|
 |  | ![tan[e+tan^(-1)(2/((e-1)^8e)-e^pi)]](/images/equations/FeigenbaumConstantApproximations/Inline49.svg) |
(21)
|
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(22)
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(23)
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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
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(24)
|
for 
,
where 
is the golden ratio, which
is good to 4 digits.
