A343393 - OEIS

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

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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).

Periodic with period 1102. - Charles R Greathouse IV, May 19 2026

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.

Index entries for linear recurrences with constant coefficients, order 552.

FORMULA

Equals 99^2/1103.

a(n) = a(n-1) - a(n-551) + a(n-552). - Charles R Greathouse IV, May 19 2026

EXAMPLE

8.88576609247506799637352674524025385312783318223...

MAPLE

evalf(99^2/1103, 120);