A318740 Decimal expansion of (sqrt((5 - sqrt(5))/2) - (sqrt(5) - 1)/2) * exp(Pi/5).

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

Offset: 1

Author

Hugo Pfoertner, Sep 16 2018

Keywords

• nonn
• cons

Comments

The second part of Ramanujan's question 352 in the Journal of the Indian Mathematical Society (IV, 40) asked "Show that 1 / (1 - exp(-Pi) / (1 + exp(-2*Pi) / (1 - exp(-3*Pi) / (1 + ...)))) = (sqrt((5 - sqrt(5))/2) - (sqrt(5) - 1)/2) * exp(Pi/5)". Also stated in Ramanujan's first letter to G. H. Hardy in 1913. Corrected version from page 28 of Berndt, Choi and Kang, see links.

Examples

			1.045077716158131508243004427816406605231289465608379931518029618...
		

Links

• B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q352, JIMS IV).
• B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q352, JIMS IV).

Crossrefs

Cf. A091667 (part 1 of question 352).

Programs

Mathematica

First[RealDigits[(Sqrt[(5 - Sqrt[5])/2] - GoldenRatio + 1)*Exp[Pi/5], 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PARI

(sqrt((1/2)*(5-sqrt(5)))-(sqrt(5)-1)/2)*exp(Pi/5)