A378960 The "tetrational mean" of 2 and 3 determined as the mutual limit of interdependent sequences.

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

Offset: 1

Author

Pham G. Hoang, Dec 12 2024

Keywords

• nonn
• cons

Comments

In an attempt to generalize the arithmetic mean (sum-based) and the geometric mean (product-based) to a similar construct for exponentiation, one can devise a simple definition using 2 interdependent sequences:

a_0 = x, b_0 = y,

a_n = exp(LambertW(log(a_{n-1}^b_{n-1}))),

b_n = exp(LambertW(log(b_{n-1}^a_{n-1}))), where x and y are the numbers for which we have to determine their "tetrational mean".

The averaging operation is the square super-root of each of the 2 possible exponentiation orders to give out the successive term of each defining sequence. The square super-root of x is exp(LambertW(log(x))) for a particular branch of the LambertW function.

If a_n and b_n converge to a number C then the "tetrational mean" of x and y is C. There may be a need to choose a particular branch of the LambertW function depending on the values of x and y (and that of log(x^y) and log(y^x)). This constant is based on the principal branch of the LambertW function.

Examples

			2.419365321844921788860745468932754354...
		

Links

• Pham G. Hoang, Table of n, a(n) for n = 1..1000
• Wikipedia, Arithmetic mean
• Wikipedia, Geometric mean
• Wikipedia, Mean
• Wikipedia, Tetration

Crossrefs

Cf. A068985, A030798, A010464.