A319331 Decimal expansion of (1 + sqrt(3 + 2*sqrt(5)))/sqrt(2) * exp(-Pi*sqrt(55)/24).

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

Offset

1,12

Comments

Ramanujan's question 489 in the Journal of the Indian Mathematical Society (V, 200) asked "Show that (1 + exp(-Pi*sqrt(55))) * (1 + exp(-3*Pi*sqrt(55))) * (1 + exp(-5*Pi*sqrt(55))) ... = (1 + sqrt(3 + 2*sqrt(5)))/sqrt(2) * exp(-Pi*sqrt(55)/24)."

Links

Table of n, a(n) for n=1..87.

B. C. Berndt, Y. S. Choi, and 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 Q489, JIMS V).

B. C. Berndt, Y. S. Choi, and 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 Q489, JIMS V).

Example

1.00000000007612269384899267384372845113345073627338819466393669...

Mathematica

RealDigits[(1 + Sqrt[3+2*Sqrt[5]])/Sqrt[2] * Exp[-Pi*Sqrt[55]/24], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

Prog

(PARI) (1+sqrt(3+2*sqrt(5)))/sqrt(2)*exp(-Pi*sqrt(55)/24)
(PARI) prodinf(k=0, (1+exp(-Pi*sqrt(55))^(2*k+1)))

Crossrefs

Sequence in context: A309700 A353823 A247444 * A371804 A010511 A073744

Adjacent sequences: A319328 A319329 A319330 * A319332 A319333 A319334

Keyword

nonn,cons

Author

Hugo Pfoertner, Sep 18 2018

Status

approved