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A384738 Decimal expansion of 3*log(2)/4 - Pi/8.

%I A384738 #18 Jun 15 2025 22:16:37
%S A384738 1,2,7,1,6,1,3,0,3,7,2,1,2,3,4,8,2,7,2,5,5,0,9,3,6,6,8,1,8,3,6,9,4,5,
%T A384738 6,5,5,3,1,9,7,8,9,2,5,8,4,8,3,0,3,2,1,2,9,6,8,6,4,1,9,3,3,0,8,1,5,6,
%U A384738 8,1,6,5,6,9,1,4,9,4,9,1,1,8,7,5,8,9,3
%N A384738 Decimal expansion of 3*log(2)/4 - Pi/8.
%C A384738 Generalization of infinite sum generating A002162 (natural logarithm of 2). That sum is Sum_{i >= 1} 1/(k*i-1) - 1/(k*i), where k = 2. Here, we set k = 4.
%H A384738 Jason Bard, <a href="/A384738/b384738.txt">Table of n, a(n) for n = 0..9999</a>
%H A384738 Steve Chow (Blackpenredpen), <a href="https://www.youtube.com/watch?v=4RfLtYDzqoc">Not telescoping: series of 1/(4n-1)-1/(4n)</a> (2019), YouTube video.
%H A384738 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 166.
%F A384738 Equals Sum_{k>=1} (1/(4k-1) - (1/4k)).
%F A384738 Equals Sum_{k>=2} zeta(k)/4^k.
%F A384738 Equals Integral_{x=1..oo} 1/(x^4+x^3+x^2+x) dx.
%F A384738 Equals A100046/2. - _Amiram Eldar_, Jun 09 2025
%e A384738 0.12716130372123482725509366818369456553197892584830...
%t A384738 RealDigits[3/4*Log[2] - Pi/8, 10, 140][[1]]
%o A384738 (PARI) 3*log(2)/4 - Pi/8 \\ _Amiram Eldar_, Jun 09 2025
%Y A384738 Cf. A002162, A100046, A384683.
%K A384738 nonn,cons
%O A384738 0,2
%A A384738 _Jason Bard_, Jun 08 2025