A078110 Decimal expansion of K210.

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

Offset

0,10

Comments

Related to modular functions and approximations to Pi: K210 is one of the most famous singular value calculated by Ramanujan. -2/sqrt(210)*log(K210/4) = 3.14159265358979323847198... agrees with Pi to 20 decimal places.

References

Lennart Berggren, Jonathan Borwein, and Peter Borwein, Pi a source Book, second edition, Springer, 2000, p. 592.

Links

Table of n, a(n) for n=0..103.

Formula

K210 = (sqrt(2)-1)^2*(2-sqrt(3))*(sqrt(7)-sqrt(6))^2*(8-3*sqrt(7))*(sqrt(10)-3)^2*(sqrt(15)-sqrt(14))*(4-sqrt(15))^2*(6-sqrt(35)).

Example

0.0000000005202524184706450480489....

Mathematica

RealDigits[(Sqrt[2]-1)^2 * (2-Sqrt[3])*(Sqrt[7]-Sqrt[6])^2 * (8-3*Sqrt[7]) * (Sqrt[10]-3)^2 * (Sqrt[15]-Sqrt[14]) * (4-Sqrt[15])^2 * (6-Sqrt[35]), 10, 120, -1][[1]] (* Amiram Eldar, Jan 16 2025 *)

Crossrefs

Sequence in context: A369286 A375694 A244813 * A334708 A201528 A093814

Adjacent sequences: A078107 A078108 A078109 * A078111 A078112 A078113

Keyword

cons,nonn

Author

Benoit Cloitre, Dec 03 2002

Status

approved