cp's OEIS Frontend

A330156 Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

%I A330156 #49 Oct 31 2024 02:03:59
%S A330156 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 A330156 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 A330156 8,2,8,8,7,0,4,3,7,1,0,9,9,5,1,3,4,5,1
%N A330156 Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].
%C A330156 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 A330156 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.4, p. 23.
%H A330156 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 A330156 Wikipedia, <a href="https://en.wikipedia.org/wiki/Continued_fraction#Continued_fraction_expansions_of_%CF%80">Continued Fraction Expansions of Pi</a>.
%H A330156 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A330156 Equals 2 / (Pi - 2).
%F A330156 Equals 1/arctan(cot(1)). - _Daniel Hoyt_, Apr 11 2020
%F A330156 From _Stefano Spezia_, Oct 26 2024: (Start)
%F A330156 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 A330156 2/(Pi - 2) = Sum_{n>=1} (2/Pi)^n (see Shamos). (End)
%F A330156 Equals A309091/2. - _Hugo Pfoertner_, Oct 28 2024
%e A330156 1.7519383938841086612039097015114538792503980068057415636404709501399828870437...
%t A330156 First[RealDigits[2/(Pi - 2), 10, 100]] (* _Paolo Xausa_, Apr 27 2024 *)
%o A330156 (PARI) 2 / (Pi - 2) \\ _Michel Marcus_, Dec 05 2019
%o A330156 (PARI) 1/atan(cotan(1)) \\ _Daniel Hoyt_, Apr 11 2020
%Y A330156 Cf. A000027, A000796, A052119, A073824, A309091.
%K A330156 nonn,cons
%O A330156 1,2
%A A330156 _Daniel Hoyt_, Dec 03 2019