A340875 - OEIS

%I #6 Jan 26 2021 10:22:28

%S 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,

%T 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,

%U 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

%N 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).

%C 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).

%C 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.

%e -1.3291223221645420016527126236974525367208157959254...

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

%K nonn,cons

%O 1,2

%A _Jon E. Schoenfield_, Jan 24 2021