A091667 Decimal expansion of ((-1-sqrt(5))/2 + sqrt((5+sqrt(5))/2))*e^((2*Pi)/5).

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

Offset

0,1

Comments

Has a nice (non-simple) continued fraction due to Ramanujan.

Continued fraction is 1/(1+q/(1+q^2/(1+q^3/(1+...)))) where q=exp(-2*Pi). - Michael Somos, Sep 12 2005

References

K. Srinivas Rao, Ramanujan, a Mathematical Genius, Eastwest Books, Chennai Madras, 2000, p. 42.

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters And Commentary, AMS, Providence RI, 1995, p. 29.

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Essays And Surveys, AMS, Providence RI, 2001, p. 243.

G. H. Hardy, Ramanujan: Twelve Lectures on subjects as suggested by his Life and Work, AMS, Chelsea Providence RI, 1999, p. 8, section 1.11.

Links

Table of n, a(n) for n=0..104.

Horst Gierhardts, Three Famous Formulas Of Ramanujan.

Srinivasa Ramanujan, Journal of the Indian Mathematical Society, Question 352 (iv, 40).

Eric Weisstein's World of Mathematics, Ramanujan Continued Fractions.

Wikipedia, Ramanujan's continued fractions.

Formula

Equals 1/A091899.

Equals exp(2*Pi/5) * A158934. - Amiram Eldar, Jan 23 2022

Example

0.998136044...

Mathematica

RealDigits[Exp[2*Pi/5]*(Sqrt[(Sqrt[5] + 5)/2] - GoldenRatio), 10, 100][[1]] (* Amiram Eldar, Jan 23 2022 *)

Prog

(PARI) {a(n)=x=exp(2/5*Pi)*(sqrt((5+sqrt(5))/2)-(1+sqrt(5))/2); floor(x*10^(n+1))%10} /* Michael Somos, Sep 12 2005 */
(PARI) {a(n)= x=exp(-2*Pi); 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 */

Crossrefs

Cf. A019694, A091899, A158934.

Sequence in context: A259149 A292825 A347058 * A300284 A347057 A334446

Adjacent sequences: A091664 A091665 A091666 * A091668 A091669 A091670

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 27 2004

Status

approved