OFFSET
0,1
COMMENTS
a(3) has 96 digits and a(4) has 479 digits.
Iterating the algebraic identity sqrt(1 + 4/x) = (1 + 2*(x + 2)/(x^2 + 3*x + 1)) * sqrt(1 + 4/(x*(x^2 + 5*x + 5)^2)) produces a rapidly converging product expansion sqrt(1 + 4/x) = product {n = 0..inf} (1 + 2*a(n)/b(n)), where a(n) and b(n) are integer sequences when x is a positive integer.
FORMULA
Let tau = 3 + 2*sqrt(2). Then a(n) = tau^(5^n) + 1/tau^(5^n).
Recurrence equation: a(n+1) = a(n)^5 - 5*a(n)^3 + 5*a(n) with initial condition a(0) = 6.
EXAMPLE
The first two terms of the product give 18 correct decimal places of sqrt(2): (1 + 2*6/29)*(1 + 2*6726/45232349) = 1.41421 35623 73095 048(5...).
CROSSREFS
KEYWORD
nonn,easy,bref
AUTHOR
Peter Bala, Nov 09 2012
STATUS
approved