cp's OEIS Frontend

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

%I A367053 #14 Nov 04 2023 06:11:37
%S A367053 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,
%T A367053 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,
%U A367053 4,4,5,1,1,2,9,1,6,2,0,2,1,1,7,5,5,6,5
%N A367053 Decimal expansion of Catalan's constant minus Serret's integral, A006752 - A102886.
%H A367053 Melissa Larson, <a href="https://www.d.umn.edu/~jgreene/masters_reports/BBP%20Paper%20final.pdf">Verifying and discovering BBP-type formulas</a>, 2008.
%H A367053 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula">Bailey-Borwein-Plouffe formula</a>.
%F A367053 Equals Integral_{x=0..1} arctan(x)/(x*(1 + x)) dx.
%F A367053 Equals Im(Polylog(2, (1 + i)/2)).
%F A367053 Equals Catalan - Pi * log(2) / 8.
%F A367053 Equals (zeta(2, 1/4) - Pi * (Pi + log(2))) / 8.
%e A367053 0.64376733288926874874201740265268236735682641173551134747577...
%p A367053 Im(polylog(2, (1 + I)/2)): evalf(%, 88);
%t A367053 First[RealDigits[Catalan - Pi * Log[2] / 8, 10, 87]]
%o A367053 (Python)  # Use a few guard digits when computing.
%o A367053 # BBP formula (1 / 16) P(2, 16, 8, (8, 8, 4, 0, -2, -2, -1, 0))
%o A367053 from decimal import Decimal as dec, getcontext
%o A367053 def BBPCatSer(n: int) -> dec:
%o A367053     getcontext().prec = n
%o A367053     s = dec(0); f = dec(1); g = dec(16)
%o A367053     for k in range(n):
%o A367053         ek = dec(8 * k)
%o A367053         s += f * ( dec(8) / (ek + 1) ** 2 + dec(8) / (ek + 2) ** 2
%o A367053                  + dec(4) / (ek + 3) ** 2 - dec(2) / (ek + 5) ** 2
%o A367053                  - dec(2) / (ek + 6) ** 2 - dec(1) / (ek + 7) ** 2 )
%o A367053         f /= g
%o A367053     return s / g
%o A367053 print(BBPCatSer(200))
%Y A367053 Cf. A006752, A102886.
%K A367053 nonn,cons
%O A367053 0,1
%A A367053 _Peter Luschny_, Nov 03 2023