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%I #32 Aug 21 2023 12:19:02
%S 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,
%T 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,
%U 2,4,6,3,7,4,4,5,2,5,1,1,6,3,5,4,8,9,2,9,9,6
%N Decimal expansion of (63/25) * (17+15*sqrt(5)) / (7+15*sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.
%C This formula that derives from Ramanujan modular equations is correct to 9 places exactly (see Ramanujan link, page 43).
%C Pi = 3.1415926535... and this approximation = 3.1415926538...
%C A quadratic number with minimal polynomial 168125x^2 - 792225x + 829521 and denominator 6725. - _Charles R Greathouse IV_, Oct 02 2022
%D Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved Jun 05 2013, (4.17) page 57.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf">Modular equations and approximations to Pi</a>, Quarterly Journal of Mathematics, XLV, 1914, p. 43.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F Equals (63/13450) * (503+75*sqrt(5)).
%F Equals the root of 829521 - 792225*x + 168125*x^2 which is > 3. - _Peter Luschny_, Nov 29 2020
%e 3.141592653805688201898390006301507822487503475774...
%p evalf((63/25)*(17+15*sqrt(5))/(7+15*sqrt(5)),100);
%t RealDigits[(63/25)*(17 + 15*Sqrt[5])/(7 + 15*Sqrt[5]), 10, 100][[1]] (* _Amiram Eldar_, Nov 29 2020 *)
%o (PARI) (63/13450) * (503+75*sqrt(5)) \\ _Michel Marcus_, Nov 29 2020
%Y Other approximations to Pi: A068028, A068079, A068089, A328927.
%K nonn,cons
%O 1,1
%A _Bernard Schott_, Nov 29 2020