A339264 Decimal expansion of (63/25) * (17+15*sqrt(5)) / (7+15*sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.

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

Offset: 1

Author

Bernard Schott, Nov 29 2020

Keywords

• nonn
• cons

Comments

This formula that derives from Ramanujan modular equations is correct to 9 places exactly (see Ramanujan link, page 43).

Pi = 3.1415926535... and this approximation = 3.1415926538...

A quadratic number with minimal polynomial 168125x^2 - 792225x + 829521 and denominator 6725. - Charles R Greathouse IV, Oct 02 2022

Examples

			3.141592653805688201898390006301507822487503475774...
		

References

• Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved Jun 05 2013, (4.17) page 57.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.

Links

• S. Ramanujan, Modular equations and approximations to Pi, Quarterly Journal of Mathematics, XLV, 1914, p. 43.
• Index entries for algebraic numbers, degree 2

Crossrefs

Other approximations to Pi: A068028, A068079, A068089, A328927.

Programs

Maple

evalf((63/25)*(17+15*sqrt(5))/(7+15*sqrt(5)),100);

Mathematica

RealDigits[(63/25)*(17 + 15*Sqrt[5])/(7 + 15*Sqrt[5]), 10, 100][[1]] (* Amiram Eldar, Nov 29 2020 *)

PARI

(63/13450) * (503+75*sqrt(5)) \\ Michel Marcus, Nov 29 2020

Formula

Equals (63/13450) * (503+75*sqrt(5)).

Equals the root of 829521 - 792225*x + 168125*x^2 which is > 3. - Peter Luschny, Nov 29 2020