A330156 - OEIS

%I #49 Oct 31 2024 02:03:59

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

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

%U 8,2,8,8,7,0,4,3,7,1,0,9,9,5,1,3,4,5,1

%N Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

%C This constant is formed from the continued fraction [1; 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...] the reciprocals of the positive integers, A000027.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.4, p. 23.

%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. 562.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Continued_fraction#Continued_fraction_expansions_of_%CF%80">Continued Fraction Expansions of Pi</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals 2 / (Pi - 2).

%F Equals 1/arctan(cot(1)). - _Daniel Hoyt_, Apr 11 2020

%F From _Stefano Spezia_, Oct 26 2024: (Start)

%F 2/(Pi - 2) = 1 + K_{n>=1} n*(n+1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 2/(Pi - 2) = 1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 + 5*6/(1 + ...))))) (see Finch at p. 23).

%F 2/(Pi - 2) = Sum_{n>=1} (2/Pi)^n (see Shamos). (End)

%F Equals A309091/2. - _Hugo Pfoertner_, Oct 28 2024

%e 1.7519383938841086612039097015114538792503980068057415636404709501399828870437...

%t First[RealDigits[2/(Pi - 2), 10, 100]] (* _Paolo Xausa_, Apr 27 2024 *)

%o (PARI) 2 / (Pi - 2) \\ _Michel Marcus_, Dec 05 2019

%o (PARI) 1/atan(cotan(1)) \\ _Daniel Hoyt_, Apr 11 2020

%Y Cf. A000027, A000796, A052119, A073824, A309091.

%K nonn,cons,changed

%O 1,2

%A _Daniel Hoyt_, Dec 03 2019