A002163 Decimal expansion of square root of 5.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

Also the limiting ratio of Lucas(n)/Fibonacci(n). - Alexander Adamchuk, Oct 10 2007

Continued fraction expansion is 2 followed by {4} repeated. - Harry J. Smith, Jun 01 2009

This is the first Lagrange number. - Alonso del Arte, Dec 06 2011

Equals Tachiya's Product_{n > 0} (1 + 2/A000032(2^n)) = 4*Product_{n > 0} (1 - 1/A000032(2^n)). - Jonathan Sondow, Jan 11 2012

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

Examples

			2.236067977499789696409173668731276235440618359611525724270897245410520...
		

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 203.
• 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

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., 22 (1968), 234-235.
• Jason Kimberley, Index of expansions of sqrt(d) in base b.
• D. Merrill, First million digits of square root of 5. [Broken link]
• Robert Nemiroff and Jerry Bonnell, The first 1 million digits of the square root of 5.
• Robert Nemiroff and Jerry Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 5.
• Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
• Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
• Y. Tachiya, Transcendence of certain infinite products, J. Number Theory 125 (2007), 182-200.
• Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, J. Int. Seq. 13 (2010) # 10.7.5, eq. (1).
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A000032, A000045, A001622.

Cf. A040002 (continued fraction).

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt(5); // Vincenzo Librandi, Feb 13 2020

Mathematica

RealDigits[N[Sqrt[5],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

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

Formula

e^(i*Pi) + 2*phi = sqrt(5).

From Christian Katzmann, Mar 19 2018: (Start)

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 2*sin(Pi/5) * 2*sin(2*Pi/5). - Gary W. Adamson, Jul 14 2022

Equals w - w^2 - w^3 + w^4 where w = exp(2*Pi*i/5). - Alexander R. Povolotsky, Nov 23 2022

From Antonio Graciá Llorente, Apr 18 2024: (Start)

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>=1} A217562(k)/A045572(k).

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)

Extensions

Sequence corrected by Paul Zimmermann, Mar 15 1996

Additional comments from Jason Earls, Mar 26 2001