The number involves the MRB constant. It is c = 0.316453098860696332954691288864309266518251...,

I've attached over 3014991 digits of it.

Here is the formula I used to compute it:

-(2*x - 3) - Log[m*Cos[x*ArcCos[3]]]/Log[-1 + Sqrt[2]] /. x -> 3014991

where m is 3014991 digits of the MRB constant.

I've found out that

  Limit[(Sqrt[2] - 1)^(-(2*x - 3) - c)/Cos[x*ArcCos[3]], x -> Infinity]= MRB constant

.!!!

As examples, it is used in the following approximation.

In[131]:= (Sqrt[2] - 
    1)^(-7.31645309886069633295469128886430926651825135500119965205926\
4044803278048652714707)/Cos[5 ArcCos[3]]

Out[131]= \
0.18785963830947022338298272934951014890322206494050103351553632625518\
3736975759412

In[132]:= (Sqrt[2] - 
    1)^(-97.3164530988606963329546912888643092665182513550011996520592\
64044803278048652714707)/Cos[50 ArcCos[3]]

Out[132]= \
0.18785964246206712024851793405427323005590309490013878617200468408947\
7231564654981

,

where the MRB constant is

0.18785964246206712024851793405427323005590309490013878617200468408947\
7231564660213703296654433107496903842345856258019061231370094759226...

I see that we have the following here: , where m is the MRB constant.

That is 2/(1 + Sqrt[2])^(3 - c) == m.

Thus That is ((m/2)(7 + 5Sqrt[2]))^(1/c) == 1 + Sqrt[2]

Attachments: