|
%I #54 Dec 16 2022 11:52:14
%S 1,1,3,1,9,7,1,7,5,3,6,7,7,4,2,0,9,6,4,3,2,4,2,7,6,9,0,6,5,4,8,9,6,4,
%T 0,0,5,0,8,7,0,4,2,4,1,7,0,2,3,9,0,4,0,8,2,3,0,4,0,7,6,1,5,2,8,2,3,6,
%U 5,0,9,1,2,5,5,6,3,9,9,6,0,7,4,5,9,9,4
%N Decimal expansion of Pi/4 + log(2)/2.
%D L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 28 (formula 154).
%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.15, p. 269.
%H Ivan Panchenko, <a href="/A231902/b231902.txt">Table of n, a(n) for n = 1..1000</a>
%H 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.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=vv85TAMIzp8">A nice integral</a>, YouTube video, 2022.
%F Equals 1 + Sum_{m>=1} -(-1)^m/(2*m*(2*m+1)) = 1 + 1/(2*3) - 1/(4*5) + 1/(6*7) - 1/(8*9) + ... .
%F From _Amiram Eldar_, Jul 16 2020: (Start)
%F Equals Integral_{x=1..oo} arctan(x)/x^2 dx.
%F Equals 1 + Integral_{x=0..1/2} log(4*x^2 + 1) dx. (End)
%F From _Bernard Schott_, Sep 07 2020: (Start)
%F Equals -Sum_{n>=1} (-1)^(n*(n+1)/2) / n [compare with A196521 formula].
%F Equals Sum_{n>=0} (32*n^2+40*n+11) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
%F Equals 1 + Sum_{k>=1} A037800(k)/(k*(k+1)) (Allouche and Shallit, 1990). - _Amiram Eldar_, Jun 01 2021
%e 1.131971753677420964324276906548964005087042417023904082304076152823650...
%t RealDigits[Pi/4 + Log[2]/2, 10, 90][[1]]
%o (PARI) default(realprecision, 100); (Pi + 2*log(2))/4 \\ _G. C. Greubel_, Aug 24 2018
%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); (Pi(R) + 2*Log(2))/4 // _G. C. Greubel_, Aug 24 2018
%Y Cf. A003881 (Pi/4), A016655 (10*(log(2)/2)), A072691 (Pi^2/12).
%Y Cf. A006752 (Catalan's constant)
%Y Cf. A196521 (Pi/4-log(2)/2).
%Y Cf. A037800.
%K nonn,cons
%O 1,3
%A _Bruno Berselli_, Nov 15 2013
|
