This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A196521 #58 Mar 07 2025 06:28:56
%S A196521 4,3,8,8,2,4,5,7,3,1,1,7,4,7,5,6,5,4,9,0,7,0,4,4,7,8,5,0,9,0,7,8,7,4,
%T A196521 3,7,0,1,1,5,4,2,2,8,2,6,6,3,6,4,8,8,2,8,1,8,3,3,9,6,1,4,3,3,3,0,2,5,
%U A196521 7,2,9,0,5,8,6
%N A196521 Decimal expansion of Pi/4-log(2)/2.
%D A196521 L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).
%H A196521 Jean-Paul Allouche and Jeffrey Shallit, <a href="https://doi.org/10.1007/BFb0097122">Sums of digits and the Hurwitz zeta function</a>, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
%F A196521 Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....
%F A196521 Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - _Peter Luschny_, Dec 06 2013
%F A196521 Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - _Robert FERREOL_, Dec 14 2015
%F A196521 Equals Integral_{x=0..1} (arctan(x)) dx = Integral_{x=0..Pi/4} (x / cos(x)^2) dx = Integral_{x=0..1/sqrt(2)} (arcsin(x)/(1-x^2)^(3/2)) dx. - _Robert FERREOL_, Dec 14 2015
%F A196521 Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - _Peter Bala_, Nov 01 2019
%F A196521 From _Bernard Schott_, Sep 07 2020: (Start)
%F A196521 Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].
%F A196521 Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
%F A196521 Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - _Amiram Eldar_, Jun 01 2021
%F A196521 From _Peter Bala_, Mar 04 2025: (Start)
%F A196521 Equals (1/2) * A342316.
%F A196521 Equals Integral_{x = 0..1} x/(x^2 - 2*x + 2) = Integral_{x = 0..1} x*(1 + x)/(2 - x^2*(1 - x)) dx.
%F A196521 Equals (5/2)*Sum_{n >= 1} 1/(n*binomial(3*n, n)*2^n). The first 10 terms of the series gives the approximate value 0.43882457311(68...), correct to 11 decimal places. (End)
%e A196521 0.438824573117475654907044785090787437011542282663648828183396143330257...
%p A196521 Digits:=100; evalf(Pi/4-log(2)/2); # _Wesley Ivan Hurt_, Dec 06 2013
%t A196521 RealDigits[Pi/4 - Log[2]/2, 10, 100] (* _Wesley Ivan Hurt_, Dec 06 2013 *)
%o A196521 (PARI) Pi/4-log(2)/2 \\ _Altug Alkan_, Dec 14 2015
%Y A196521 Cf. A003881, A016655 (10*log(2)/2), A033264.
%Y A196521 Cf. A231902 (Pi/4+log(2)/2), A342316.
%K A196521 nonn,cons,easy
%O A196521 0,1
%A A196521 _R. J. Mathar_, Oct 03 2011