A318739 Decimal expansion of Pi^2 / 24 - (1/12) * log(2 + sqrt(5))^2.

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

Offset

0,1

Comments

Ramanujan's question 606 in the Journal of the Indian Mathematical Society (VI, 239) asked "Show that Sum_{n>=0} (sqrt(5) - 2)^(2*n + 1) / (2*n + 1)^2 = Pi^2/24 - (1/12) * (log(2 + sqrt(5)))^2".

Links

Table of n, a(n) for n=0..85.

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 Q606, JIMS VI).

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 Q606, JIMS VI).

Example

0.2375599012791608147454069882378567272924327127647254563224136358251...

Mathematica

RealDigits[Pi^2/24 - Log[2+Sqrt[5]]^2/12, 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

Prog

(PARI) Pi^2/24-(1/12)*log(2+sqrt(5))^2
(PARI) suminf(k=0, (sqrt(5)-2)^(2*k+1)/(2*k+1)^2)

Crossrefs

Sequence in context: A329412 A335292 A060940 * A205299 A222242 A069771

Adjacent sequences: A318736 A318737 A318738 * A318740 A318741 A318742

Keyword

nonn,cons

Author

Hugo Pfoertner, Sep 16 2018

Status

approved