New breakthrough in understanding (numerically) the MRB constant

Ideas
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Here is my latest attempt to find a super-fast method of computing the MRB constat as of 11/18/2014. (I still have a ways to go.) See Try to beat these MRB constant records! for my attempts at computing it. The notebook is attached so you can copy and edit it.

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Reconstruct and greatly expand the above RootApproximant's (here f) and find they are approximated well by certain
 ((1+Sqrt[2])^n+(1-Sqrt[2])^n)/2.

Clear[key, f]

Block[{$MaxExtraPrecision = 1000}, For[x = 1, x <= 48,
        key[x] = Table[Convergents[Sqrt[2], 100][[2 n]], {n, 50}][[-x]];
        f[x] = Numerator[key[x]] - Denominator[key[x]] Sqrt[2];              
  Print[N[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2 /. n -> 100 + 2 (x - 1), 
     120 + x] - N[f[x], 120 + x]*ChebyshevT[100, 3]];
  x++]]


2.6387695903457195648070395905136485002528815*10^-39

4.527412857479332018505153596026307976091567*10^-40

7.767812414187964629605256710213628540205842*10^-41

1.332745910334467592580004301018691480319384*10^-41

2.28663047818840925874769095898520341710461*10^-42

3.9232376578577962668610274372430569943380*10^-43

6.7312116526268501368925503360630779498198*10^-44

1.1548933371831381527450276439478977555385*10^-44

1.981483704719787795776155276243085834114*10^-45

3.39968856487345247206655217979537449296*10^-46

5.83294342042836874637760316341388616634*10^-47

1.00077487383568775760009718252957206843*10^-47

1.7170582258575779922297993176354624423*10^-48

2.946006167885903773778240805170539698*10^-49

5.054547487396427203714516546686137636*10^-50

8.67223245519525484504691228411428839*10^-51

1.48791985720725703313630823782435395*10^-51

2.5528668804828735377093714283183531*10^-52

4.380027108246708948931461916665789*10^-53

7.514938446515183164950572168112005*10^-54

1.289359596624009500388813842014146*10^-54

2.21219133228873837382310883972869*10^-55

3.7955202749233523905051461823070*10^-56

6.5120832665273060479978869655528*10^-57

1.1172968499303123829358599702467*10^-57

1.916978330545682496172728559273*10^-58

3.28901483970971147677771653172*10^-59

5.64305732801443898939013597563*10^-60

9.6819557098951916856365053661*10^-61

1.6611609792267602199176724406*10^-61

2.850101654653696338695292773*10^-62

4.89000135654575832995032232*10^-63

8.38991592737586592749006203*10^-64

1.43948199879761226543714897*10^-64

2.4697606540980766513283182*10^-65

4.237439366123372535984192*10^-66

7.270296557594687026219692*10^-67

1.247385684334396797476231*10^-67

2.14017548411693758637696*10^-68

3.6719606135765754349947*10^-69

6.3000884029007674619849*10^-70

1.0809242816388504219628*10^-70

1.854572869323350697919*10^-71

3.18194399551599967885*10^-72

5.45935279862491093918*10^-73

9.3667683658946884656*10^-74

1.6070822091190214020*10^-74

2.757248888194399464*10^-75
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