A364945 - OEIS

%I #23 Nov 15 2024 06:59:15

%S 0,8,4,0,3,4,4,0,5,8,2,2,7,8,0,9,8,4,9,4,5,3,9,6,4,8,5,0,6,7,6,1,5,8,

%T 8,9,2,2,5,8,5,0,6,2,5,7,1,8,3,2,7,8,6,5,7,3,3,5,0,1,8,8,0,3,7,8,2,3,

%U 6,9,8,0,2,2,3,7,4,5,2,3,0,5,2,0,6,4,3,4,8

%N Decimal expansion of 1-Catalan.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 56.

%H J. W. L. Glaisher, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.%24b416778&amp;seq=60">Numerical Values of the Series 1 - 1/3^n + 1/5^n - 1/7^n + 1/9^n - &c. For n = 2, 4, 6</a>, Messenger of Mathematics, Vol. 42 (1913), pp. 50-58. [Available only in the USA through the Hathi Trust]

%H 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. 122.

%F Equals 3F2( -1/2,-1/2,1 ; 1/2,1/2 ; -1) = (1/4) * Sum_{k>=0} (-1)^k/(k-1/2)^2.

%F Equals 1-A006752.

%F Equals Sum_{k>=1} k*zeta(2*k+1)/16^k (Glaisher, 1913). - _Amiram Eldar_, Aug 14 2023

%F Equals Sum_{k>=1} 16*k/(16*k^2 - 1)^2 = Integral_{x=1..oo} log(x)/(x^4 + x^2) dx (see Shamos). - _Stefano Spezia_, Nov 14 2024

%e 0.084034405822780984945396...

%p evalf(1-Catalan) ;

%t RealDigits[1 - Catalan, 10, 120, -1][[1]] (* _Amiram Eldar_, Aug 14 2023 *)

%o (PARI) 1 - Catalan \\ _Michel Marcus_, Aug 14 2023

%Y Cf. A006752.

%K nonn,cons

%O 0,2

%A _R. J. Mathar_, Aug 14 2023