A343393 Decimal expansion of 99^2/1103, an approximation to 2*Pi*sqrt(2) from Ramanujan.

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

Offset

1,1

Comments

Srinivasa Ramanujan produced this curious approximation to 2*Pi*sqrt(2) (A343392) with dividing 99^2 by prime 1103 (see link Prime Curios!). This approximation comes from the 1st term of the series (44) page 47 at the Ramanujan link.

This formula is correct to 5 places exactly with 2*Pi*sqrt(2) = 8.885765... while 99^2/1103 = 8.885766...

Indeed, in the Ramanujan paper, there is 1/(2*Pi*sqrt(2)) = 1103/99^2 + ..., and in the case of these two numbers, the approximation becomes correct to 8 places exactly with 1/(2*Pi*sqrt(2)) = 0.112539539... while 1103/99^2 = 0.112539536... (see David Wells).

References

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 3.14159..., page 36.

Links

Table of n, a(n) for n=1..94.

Chris K. Caldwell and G. L. Honaker, Jr., 1103, 1st comment, Prime Curios!

S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 47.

Formula

Equals 99^2/1103.

Example

8.88576609247506799637352674524025385312783318223...

Maple

evalf(99^2/1103, 120);

Mathematica

RealDigits[99^2/1103, 10, 100][[1]] (* Amiram Eldar, Apr 13 2021 *)

Crossrefs

Cf. A343392.

Sequence in context: A023412 A336211 A343392 * A023413 A188727 A361011

Adjacent sequences: A343390 A343391 A343392 * A343394 A343395 A343396

Keyword

nonn,cons

Author

Bernard Schott, Apr 13 2021

Status

approved