cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A352548 Decimal expansion of 22*Pi^4.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Jun 21 2022

Keywords

Comments

Gardner (1985) wrote: "A more astounding discovery is that: 22 pi^4 = 2143. A few multiplications, and the 10 million-plus decimals of pi have vanished. (Can this remarkable relationship mirror some as yet undiscovered facet of physical reality?)" In the Postscript to the 1999 reprint (cf. links) he writes "Divide (...) 2143 by 22 and hit the square-root button twice. You will get pi to eight decimals", and credits this discovery to Srinivasa Ramanujan. The MathOverflow page also mentions this and the near-integer 10*Pi^4 - 1/11 ≈ 974.0000012.
Even after a(0..4) = 0, the digits '0' and '1' remain significantly more frequent than other digits: almost 3 times more frequent than the digit 3 within the first 100 terms, and still 30 - 40 percent more frequent than half of the other digits among the first 1000 terms. However, we don't consider that to be a "secret hidden in pi".

References

  • Martin Gardner, "Slicing Pi into Millions", Discover, 6:50, January 1985.

Links

Crossrefs

Cf. A000796 (decimal digits of Pi), A328927 (decimal digits of (2143/22)^1/4).

Programs

  • Mathematica
    RealDigits[22*Pi^4, 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)
  • PARI
    A352548_first(N)=localprec(N+5);digits(22*Pi^4\10^(4-N)) \\ First N terms of this sequence, i.e., a(4 .. 5-N).

Formula

22*Pi^4 = 2143.000002748053619201687319151512447494006884799...

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

Original entry on oeis.org

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

Views

Author

Bernard Schott, Nov 29 2020

Keywords

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

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
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.