|
%I #26 Jun 08 2023 01:57:40
%S 9,9,9,9,9,9,2,0,8,7,3,2,9,0,0,7,9,3,1,2,7,4,7,3,0,4,0,9,3,3,7,1,5,7,
%T 8,6,5,1,5,1,5,9,4,1,5,0,0,5,4,0,9,4,7,8,9,4,4,7,8,4,1,2,5,3,6,9,9,2,
%U 1,5,6,7,5,7,8,5,0,4,2,0,6,3,9,3,3,5,7,4,4,3,0,4,8,1,1,0,7,9,9,3,8,4,8,8,7
%N Decimal expansion of exp(2*Pi/v) * (v/(1+(5^(3/4)/((1+v)/2)^(5/2)-1)^(1/5))-(1+v)/2), where v = sqrt(5).
%C Has a nice (non-simple) continued fraction due to Ramanujan.
%C Continued fraction is 1/(1+q/(1+q^2/(1+q^3/(1+...)))) where q=exp(-2*Pi*sqrt(5)). - _Michael Somos_, Sep 12 2005
%D G. H. Hardy, Ramanujan: Twelve Lectures on subjects as suggested by his Life and Work, AMS Chelsea Providence RI, 1999, p. 8, section (1.12).
%D K. S. Rao, Srinivasa Ramanujan, a Mathematical Genius, Eastwest Books Chennai Madras, 2000, p. 42.
%H I. E. S. Cartuja, <a href="https://web.archive.org/web/20220621081135/https://thales.cica.es/rd/Recursos/rd97/Biografias/07-1-b-r.html">Srinivasa Ramanujan</a> (text in Spanish). [Wayback Machine copy]
%H Horst Gierhardts, <a href="http://www.gierhardt.de/mathematik/ramaformeln.html">Three Famous Formulas of Ramanujan</a>.
%H Shriram Sarvotham, <a href="https://web.archive.org/web/20211019180856/https://www.ece.rice.edu/~shri/ramanujan.html">Srinivasa Ramanujan</a>. [Wayback Machine copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanContinuedFractions.html">Ramanujan Continued Fractions</a>.
%F Equals 1/A091900.
%e 0.99999920873290079312747304093371578651515941500540...
%t With[{v = Sqrt[5]}, RealDigits[Exp[2*Pi/v] * (v/(1 + (5^(3/4)/((1 + v)/2)^(5/2) - 1)^(1/5)) - (1 + v)/2), 10, 120][[1]]] (* _Amiram Eldar_, Jun 08 2023 *)
%o (PARI) {a(n)=local(s); s=sqrt(5); x=exp(2*Pi/s)*(s/(1+(5^(3/4)/((1+s)/2)^(5/2)-1)^(1/5))-(1+s)/2); floor(x*10^(n+1))%10} /* _Michael Somos_, Sep 12 2005 */
%o (PARI) {a(n)= x=exp(-2*Pi*sqrt(5)); x=contfracpnqn(matrix(2,oo,i,j,if(j==1,i==1,if(i==1,x,1)^(j-2)))); x=t[1,1]/t[2,1]; floor(x*10^(n+1))%10} /* _Michael Somos_, Sep 12 2005 */
%Y Cf. A091900.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Jan 27 2004
|
