fun constants 1: s

s=1.776775040097054697479730744038756748637… is the unique
nontrivial real root of s+1 = s^s, and it’s got some fun properties. It is sort of like an exponential analogue of the golden mean, but not in the same way that the omega constant is.

(s+1)^{s+1} = (s^{s})^{s^{s}} = s^{s^{s+1}} = s^{s^{s^{s}}}

(s+1)s = s^2 + s
And also:
(s+1)s = s\cdot s^{s} = s^{s+1} = s^{s^{s}}
so:
s^2 + s = s^{s^{s}}

The import of this is that we can find interesting additive decompositions of tetrational towers of {}^{n}s on some occasions. I think:
{}^n(s+1) = {}^{2n}s

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