A277435 Decimal expansion of lim_{n->inf} (2 - sqrt(2)^^n)/log(2)^n, where x^^n denotes tetration.

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

Offset: 0

Author

Vladimir Reshetnikov, Oct 14 2016

Keywords

• nonn
• cons

Comments

Tetration x^^n is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). Note that sqrt(2)^^inf = lim_{n->inf} sqrt(2)^^n = 2. Asymptotically, sqrt(2)^^n = 2 - O(log(2)^n). This constant is the coefficient in the O(log(2)^n) term. Furthermore, sqrt(2)^^n = 2 - a*log(2)^n + (a^2/(4*(1 - 1/log(2))))*log(2)^(2*n) + O(log(2)^(3*n)).

Examples

			0.63209866105082925035545064599078...
		

Links

• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Tetration

Crossrefs

Cf. A198094, A260691.

Programs

Mathematica

RealDigits[SequenceLimit[1`200 Table[(2 - Power @@ Table[Sqrt[2], {n}])/Log[2]^n, {n, 1, 200}]], 10, 100][[1]]

Formula

a = 2*sqrt(2)*A260691/(1-log(2)).