A091668 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).

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, 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, 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

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*sqrt(5)). - Michael Somos, Sep 12 2005

References

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).

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

Links

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

I. E. S. Cartuja, Srinivasa Ramanujan (text in Spanish). [Wayback Machine copy]

Horst Gierhardts, Three Famous Formulas of Ramanujan.

Shriram Sarvotham, Srinivasa Ramanujan. [Wayback Machine copy]

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

Formula

Equals 1/A091900.

Example

0.99999920873290079312747304093371578651515941500540...

Mathematica

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 *)

Prog

(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 */
(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 */

Crossrefs

Cf. A091900.

Sequence in context: A346435 A346449 A290665 * A196498 A277535 A292905

Adjacent sequences: A091665 A091666 A091667 * A091669 A091670 A091671

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 27 2004

Status

approved