A340875 Decimal expansion of lim_{k->infinity} k - 1/x(k) + log(x(k)) where x(k) is the real number from which k "add the square" iterations reach exactly 1 (negated).

1, 3, 2, 9, 1, 2, 2, 3, 2, 2, 1, 6, 4, 5, 4, 2, 0, 0, 1, 6, 5, 2, 7, 1, 2, 6, 2, 3, 6, 9, 7, 4, 5, 2, 5, 3, 6, 7, 2, 0, 8, 1, 5, 7, 9, 5, 9, 2, 5, 4, 2, 8, 0, 1, 7, 3, 7, 8, 3, 8, 8, 0, 7, 5, 2, 2, 4, 2, 7, 2, 3, 8, 4, 0, 3, 0, 8, 0, 1, 5, 4, 3, 2, 5, 9, 1, 5, 4, 1, 8, 0, 8, 0, 4, 9, 5, 5, 5, 7, 2

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.

Links

Table of n, a(n) for n=1..100.

Example

-1.3291223221645420016527126236974525367208157959254...

Crossrefs

Cf. A340745, A340824, A340825, A340844, A340845.

Sequence in context: A060481 A335228 A228492 * A329038 A272491 A010271

Adjacent sequences: A340872 A340873 A340874 * A340876 A340877 A340878

Keyword

nonn,cons

Author

Jon E. Schoenfield, Jan 24 2021

Status

approved