A367053 Decimal expansion of Catalan's constant minus Serret's integral, A006752 - A102886.

6, 4, 3, 7, 6, 7, 3, 3, 2, 8, 8, 9, 2, 6, 8, 7, 4, 8, 7, 4, 2, 0, 1, 7, 4, 0, 2, 6, 5, 2, 6, 8, 2, 3, 6, 7, 3, 5, 6, 8, 2, 6, 4, 1, 1, 7, 3, 5, 5, 1, 1, 3, 4, 7, 4, 7, 5, 7, 7, 3, 7, 1, 2, 9, 7, 2, 4, 7, 4, 4, 5, 1, 1, 2, 9, 1, 6, 2, 0, 2, 1, 1, 7, 5, 5, 6, 5

Offset: 0

Author

Peter Luschny, Nov 03 2023

Keywords

• nonn
• cons

Examples

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Links

• Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
• Wikipedia, Bailey-Borwein-Plouffe formula.

Crossrefs

Cf. A006752, A102886.

Programs

Maple

Im(polylog(2, (1 + I)/2)): evalf(%, 88);

Mathematica

First[RealDigits[Catalan - Pi * Log[2] / 8, 10, 87]]

Python

# Use a few guard digits when computing.
# BBP formula (1 / 16) P(2, 16, 8, (8, 8, 4, 0, -2, -2, -1, 0))
from decimal import Decimal as dec, getcontext
def BBPCatSer(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(16)
    for k in range(n):
        ek = dec(8 * k)
        s += f * ( dec(8) / (ek + 1) ** 2 + dec(8) / (ek + 2) ** 2
                 + dec(4) / (ek + 3) ** 2 - dec(2) / (ek + 5) ** 2
                 - dec(2) / (ek + 6) ** 2 - dec(1) / (ek + 7) ** 2 )
        f /= g
    return s / g
print(BBPCatSer(200))

Formula

Equals Integral_{x=0..1} arctan(x)/(x*(1 + x)) dx.

Equals Im(Polylog(2, (1 + i)/2)).

Equals Catalan - Pi * log(2) / 8.

Equals (zeta(2, 1/4) - Pi * (Pi + log(2))) / 8.