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

0,1

References

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 266, eq. (4.183).

Links

Table of n, a(n) for n=0..86.

Melissa Larson, Verifying and discovering BBP-type formulas, M.Sc. Thesis, University of Minnesota Duluth, 2008.

Wikipedia, Bailey-Borwein-Plouffe formula.

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.

Equals Sum_{k>=1} (-1)^(k+1)*H(2*k-1)/(2*k-1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Olaikhan, 2021). - Amiram Eldar, Feb 03 2026

Example

0.64376733288926874874201740265268236735682641173551134747577...

Maple

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

Mathematica

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

Prog

(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))
(PARI) Catalan - Pi*log(2)/8 \\ Amiram Eldar, Feb 03 2026

Crossrefs

Cf. A001008, A002805, A006752, A102886.

Sequence in context: A079624 A035335 A011097 * A195475 A321786 A188725

Adjacent sequences: A367050 A367051 A367052 * A367054 A367055 A367056

Keyword

nonn,cons

Author

Peter Luschny, Nov 03 2023

Status

approved