Infinite tetration giving I*Omega costant=I*0.567143...

[bo198214]

one can see numerically that in the limit \( \theta_n = \alpha+n \) for some constant \( \alpha \).

As far as I did it (1850 terms), nothing is constant-You mean you replaced small difference of sin from -1 with an argument?
But as sin argument nears n*(3pi/2), can You do it?

As I see it since teta(n+1)-teta(n) = 1 when n-> infinity, there shall be no differences between teta(n) and n integer part.

There can not be one also in reals, since limit n->infinity is 1.

So this constant seems suspicious to me-or may be I misunderstood something.

Whats not clear about this?
Just compute \( \theta_n - (\alpha + n) \) with \( \alpha=\frac{1}{2}-\frac{1}{2\pi}\arcsin\left(\frac{1-A}{B}\right)-2 \). You see numerically that \( \theta_n - (\alpha + n)\to 0 \) and that means that \( \theta_n = \alpha+n \) in the limit.