OFFSET
1,2
COMMENTS
Define the real-valued sequence {x(0), x(1), x(2), ...} such that x(k) = (sqrt(4*x(k-1) + 1) - 1)/2 for k > 0 and x(0)=1. Then x(k-1) = x(k) + x(k)^2 for k > 0. In other words, if we iterate the map x -> x + x^2 starting at x = x(k), the k-th iteration will bring us to exactly 1. If we start at any value x such that x(k) < x < x(k-1), it will require k iterations to reach an x value greater than 1 (cf. A340745).
For large values of k, k approaches 1/x(k) - log(x(k)) + c0 + (1/2)*x(k) - (1/3)*x(k)^2 + ... (see A340825), where c0 = -1.329122322... is the constant whose decimal expansion is this sequence.
EXAMPLE
-1.3291223221645420016527126236974525367208157959254...
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jon E. Schoenfield, Jan 24 2021
STATUS
approved