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

%I A339264 #32 Aug 21 2023 12:19:02
%S A339264 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 A339264 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 A339264 2,4,6,3,7,4,4,5,2,5,1,1,6,3,5,4,8,9,2,9,9,6
%N A339264 Decimal expansion of (63/25) * (17+15*sqrt(5)) / (7+15*sqrt(5)): an approximation for Pi from Srinivasa Ramanujan.
%C A339264 This formula that derives from Ramanujan modular equations is correct to 9 places exactly (see Ramanujan link, page 43).
%C A339264 Pi = 3.1415926535... and this approximation = 3.1415926538...
%C A339264 A quadratic number with minimal polynomial 168125x^2 - 792225x + 829521 and denominator 6725. - _Charles R Greathouse IV_, Oct 02 2022
%D A339264 Jörg Arndt and Christoph Haenel, Pi Unleashed, Springer-Verlag, 2006, retrieved Jun 05 2013, (4.17) page 57.
%D A339264 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 3.14159 (Pi), page 36.
%H A339264 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 A339264 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F A339264 Equals (63/13450) * (503+75*sqrt(5)).
%F A339264 Equals the root of 829521 - 792225*x + 168125*x^2 which is > 3. - _Peter Luschny_, Nov 29 2020
%e A339264 3.141592653805688201898390006301507822487503475774...
%p A339264 evalf((63/25)*(17+15*sqrt(5))/(7+15*sqrt(5)),100);
%t A339264 RealDigits[(63/25)*(17 + 15*Sqrt[5])/(7 + 15*Sqrt[5]), 10, 100][[1]] (* _Amiram Eldar_, Nov 29 2020 *)
%o A339264 (PARI) (63/13450) * (503+75*sqrt(5)) \\ _Michel Marcus_, Nov 29 2020
%Y A339264 Other approximations to Pi: A068028, A068079, A068089, A328927.
%K A339264 nonn,cons
%O A339264 1,1
%A A339264 _Bernard Schott_, Nov 29 2020