[From Simon Plouffe: https://vixra.org/pdf/1408.0181v1.pdf ]
In[]:=
pl[n_]:=1+Sqrt[24^n+2Sqrt[16^n+1]]/(2^(2n+2))
In[]:=
ListLinePlot[Accumulate[(-1)^First[RealDigits[Sqrt[2],2,10000]]]]
Out[]=
2000
4000
6000
8000
10000
-50
50
In[]:=
pl[100]
Out[]=
1+(3213876088517980551083924184682325205044405987565585670602752+2√2582249878086908589655919172003011874329705792829223512830659356540647622016841194629645353280137831435903171972747493377)6427752177035961102167848369364650410088811975131171341205504
In[]:=
ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[100],2,10000]]]]
Out[]=
2000
4000
6000
8000
10000
100
200
300
400
500
In[]:=
ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[100],2,100000]]]]
Out[]=
20000
40000
60000
80000
100000
-200
200
400
In[]:=
ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[50],2,100000]]]]
Out[]=
20000
40000
60000
80000
100000
-100
100
200
300
In[]:=
ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[150],2,100000]]]]
Out[]=
20000
40000
60000
80000
100000
200
400
600
In[]:=
ArrayPlot[Partition[First[RealDigits[pl[50],2,10000]],100]]
Out[]=
In[]:=
Length/@Split[First[RealDigits[pl[50],2,10000]]]
Out[]=
In[]:=
ListStepPlot[Length/@Split[First[RealDigits[pl[50],2,20000]]],PlotRangeAll]
Out[]=
1000
2000
3000
4000
5000
6000
7000
50
100
150
200
In[]:=
pl[n_]:=1+Sqrt[24^n+2Sqrt[16^n+1]]/(2^(2n+2))
In[]:=
pl2[n_]:=1+Sqrt[2^n+2Sqrt[2^(2n-2)+1]]/2^(n+1)
In[]:=
ListStepPlot[Length/@Split[First[RealDigits[pl2[50],2,10000]]],PlotRangeAll]
Out[]=
1000
2000
3000
4000
5000
5
10
15
20
25
In[]:=
ListStepPlot[Length/@Split[First[RealDigits[pl2[51],2,10000]]],PlotRangeAll]
Out[]=
1000
2000
3000
4000
20
40
60
80
100
In[]:=
ListStepPlot[Length/@Split[First[RealDigits[pl2[51],2,2000]]],PlotRangeAll]
https://www.math.stonybrook.edu/preprints/ims99-3.pdf
Even exponents correspond to Sqrt[2^n+1] ≈ Sqrt[2^n] which is 2^(n/2) for even n, and a mess for odd n.

Simpler case: