A007493 Decimal expansion of Wallis's number, the real root of x^3 - 2*x - 5. (Formerly M0036)

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

Offset

1,1

Comments

"The real solution to the equation x^3 - 2x - 5 = 0. This equation was solved by [the English mathematician John] Wallis [1616-1703] to illustrate Newton's method for the numerical solution of equations.

"It has since served as a test for many subsequent methods of approximation and its real root is now known to 4000 digits." [Gruenberger]

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. E. Smith, A Source Book in Mathematics, McGraw-Hill, 1929, pp. 247-248.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 27.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

Fred Gruenberger, How to handle numbers with thousands of digits, and why one might want to, Computer Recreations, Scientific American, 250 (No. 4, 1984), 19-26.

W. G. Horner, A new method of solving numerical equations of all orders, by continuous approximation, Phil. Trans. Royal Soc., 1819, pp. 308-335.

D. Olivastro, Ancient Puzzles, Bantam Books, NY (1993), cover page and pp. 58-59. (Annotated scanned copy)

Eric Weisstein's World of Mathematics, Wallis's Constant

Formula

Equals (5/2 - sqrt(643/108))^(1/3) + (5/2 + sqrt(643/108))^(1/3). - Michal Paulovic, Mar 19 2023

Example

2.094551481542326591482386540579302963857306105628239180304128529...

Mathematica

RealDigits[ N[ 1/3* (135/2 - (3*Sqrt[1929])/2)^(1/3) + (1/2*(45 + Sqrt[1929]) )^(1/3) / 3^(2/3), 100]][[1]]

Prog

(PARI) { default(realprecision, 20080); x=NULL; p=x^3 - 2*x - 5; rs=polroots(p); r=real(rs[1]); for (n=1, 20000, d=floor(r); r=(r-d)*10; write("b007493.txt", n, " ", d)); } \\ Harry J. Smith, May 03 2009
(PARI) polrootsreal(x^3-2*x-5)[1] \\ Charles R Greathouse IV, Apr 14 2014

Crossrefs

Cf. A058297 Continued fraction.

Sequence in context: A343882 A168229 A019693 * A262177 A136319 A176057

Adjacent sequences: A007490 A007491 A007492 * A007494 A007495 A007496

Keyword

cons,nonn

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

Status

approved