Paris Constant

The golden ratio can be written in terms of a nested radical in the beautiful form

(1)

which can be written recursively as

(2)

for , with .

Paris (1987) proved approaches at a constant rate, namely

(3)

as , where

(4)

(OEIS A105415) is the Paris constant.

A product formula for is given by

(5)

(Finch 2003, p. 8).

Another formula is given by letting be the analytic solution to the functional equation

(6)

for , subject to initial conditions and . Then

(7)

(Finch 2003, p. 8).

A close approximation is , which is good to 4 decimal places (M. Stark, pers. comm.).