[From Simon Plouffe: https://vixra.org/pdf/1408.0181v1.pdf ]
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pl[n_]:=1+Sqrt[24^n+2Sqrt[16^n+1]]/(2^(2n+2))
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ListLinePlot[Accumulate[(-1)^First[RealDigits[Sqrt[2],2,10000]]]]
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pl[100]
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1+(3213876088517980551083924184682325205044405987565585670602752+2√2582249878086908589655919172003011874329705792829223512830659356540647622016841194629645353280137831435903171972747493377)6427752177035961102167848369364650410088811975131171341205504
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ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[100],2,10000]]]]
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ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[100],2,100000]]]]
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ListLinePlot[Accumulate[(-1)^First[RealDigits[pl[50],2,100000]]]]
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