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

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

Author

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

Keywords

• cons
• nonn

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]

Examples

			2.094551481542326591482386540579302963857306105628239180304128529...
		

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
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A058297 (continued fraction).

Programs

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]]

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

Formula

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

Extensions

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