|
|
A002163
|
|
Decimal expansion of square root of 5.
(Formerly M0293 N0105)
|
|
78
|
|
|
2, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, 7, 8, 9, 6, 9, 6, 4, 0, 9, 1, 7, 3, 6, 6, 8, 7, 3, 1, 2, 7, 6, 2, 3, 5, 4, 4, 0, 6, 1, 8, 3, 5, 9, 6, 1, 1, 5, 2, 5, 7, 2, 4, 2, 7, 0, 8, 9, 7, 2, 4, 5, 4, 1, 0, 5, 2, 0, 9, 2, 5, 6, 3, 7, 8, 0, 4, 8, 9, 9, 4, 1, 4, 4, 1, 4, 4, 0, 8, 3, 7, 8, 7, 8, 2, 2, 7
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Continued fraction expansion is 2 followed by {4} repeated. - Harry J. Smith, Jun 01 2009
A computation similar, with that of the universal parabolic constant, performed on the curve cosh(x) with the parameters of the osculating parabola, gives as result 2*sinh(arccosh(3/2)), that is sqrt(5) instead of 2.2955871... for the parabola. - Jean-François Alcover, Jul 18 2013
Because sqrt(5) = -1 + 2*phi, with the golden section phi from A001622, this is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Jan 08 2018
This constant appears in the theorem of Hurwitz on the best approximation of any irrational number with infinitely many rationals: |theta - h/k| < 1/(sqrt(5)*k^2). See Niven, also for the Hurwitz 1891 reference. - Wolfdieter Lang, May 27 2018
Diameter of a sphere whose surface area equals 5*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018
|
|
REFERENCES
|
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, Theorem 1.5, pp. 6, 14.
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 106.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.
|
|
LINKS
|
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
|
|
FORMULA
|
e^(i*Pi) + 2*phi = sqrt(5).
Equals Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)).
Equals Sum_{n>=0} 25*(2*n+1)!/(n!^2*3^(2*n+3)). (End)
Equals -1 + 2*phi, with phi = A001622. An integer number in the real quadratic number field Q(sqrt(5)). - Wolfdieter Lang, May 09 2018
Equals Sum_{k>=0} binomial(2*k,k)/5^k. - Amiram Eldar, Aug 03 2020
Equals Product_{k>=0} ((10*k + 2)(10*k + 4)(10*k + 6)(10*k + 8))/((10*k + 1)*(10*k + 3)*(10*k + 7)*(10*k + 9)).
Equals Product_{k>=0} (1/2)*(((4*k + 9)/(4*k + 1))^(1/2) + ((4*k + 1)/(4*k + 9))^(1/2)).
Equals Product_{k>=1} (phi^k + phi)/(phi^k + phi - 1), with phi = A001622.
Equals Product_{k>=0} (Fibonacci(2*k + 3) + (-1)^k)/(Fibonacci(2*k + 3) - (-1)^k). (End)
|
|
EXAMPLE
|
2.236067977499789696409173668731276235440618359611525724270897245410520...
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) default(realprecision, 20080); x=sqrt(5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002163.txt", n, " ", d)); \\ Harry J. Smith, Jun 01 2009
(Magma) SetDefaultRealField(RealField(100)); Sqrt(5); // Vincenzo Librandi, Feb 13 2020
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|