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A002163 Decimal expansion of square root of 5.

%I A002163 M0293 N0105 #143 Mar 16 2025 08:18:26
%S A002163 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,
%T A002163 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,
%U A002163 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
%N A002163 Decimal expansion of square root of 5.
%C A002163 Also the limiting ratio of Lucas(n)/Fibonacci(n). - _Alexander Adamchuk_, Oct 10 2007
%C A002163 Continued fraction expansion is 2 followed by {4} repeated. - _Harry J. Smith_, Jun 01 2009
%C A002163 This is the first Lagrange number. - _Alonso del Arte_, Dec 06 2011
%C A002163 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
%C A002163 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
%C A002163 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
%C A002163 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
%C A002163 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
%D A002163 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 203.
%D A002163 W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
%D A002163 Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963, Theorem 1.5, pp. 6, 14.
%D A002163 Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 106.
%D A002163 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002163 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002163 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.
%H A002163 Harry J. Smith, <a href="/A002163/b002163.txt">Table of n, a(n) for n = 1..20000</a>
%H A002163 M. F. Jones, <a href="http://www.jstor.org/stable/2004806">22900D approximations to the square roots of the primes less than 100</a>, Math. Comp., 22 (1968), 234-235.
%H A002163 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/sqrt_base">Index of expansions of sqrt(d) in base b</a>.
%H A002163 D. Merrill, <a href="http://www.netcom.com/~merrills/sqrt51000000.txt">First million digits of square root of 5</a>. [Broken link]
%H A002163 Robert Nemiroff and Jerry Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt5.1mil">The first 1 million digits of the square root of 5</a>.
%H A002163 Robert Nemiroff and Jerry Bonnell, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/sqrt5.txt">The first 1 million digits of the square root of 5</a>.
%H A002163 Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0983.00008&amp;format=complete">Zentralblatt review</a>.
%H A002163 Jonathan Sondow, <a href="http://arxiv.org/abs/1106.4246">Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers</a>, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
%H A002163 Y. Tachiya, <a href="http://dx.doi.org/10.1016/j.jnt.2006.11.006">Transcendence of certain infinite products</a>, J. Number Theory 125 (2007), 182-200.
%H A002163 Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula2/witula40r.html">Ramanujan Cubic Polynomials of the Second Kind</a>, J. Int. Seq. 13 (2010) # 10.7.5, eq. (1).
%H A002163 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A002163 e^(i*Pi) + 2*phi = sqrt(5).
%F A002163 From _Christian Katzmann_, Mar 19 2018: (Start)
%F A002163 Equals Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)).
%F A002163 Equals Sum_{n>=0} 25*(2*n+1)!/(n!^2*3^(2*n+3)). (End)
%F A002163 Equals -1 + 2*phi, with phi = A001622. An integer number in the real quadratic number field Q(sqrt(5)). - _Wolfdieter Lang_, May 09 2018
%F A002163 Equals Sum_{k>=0} binomial(2*k,k)/5^k. - _Amiram Eldar_, Aug 03 2020
%F A002163 Equals 2*sin(Pi/5) * 2*sin(2*Pi/5). - _Gary W. Adamson_, Jul 14 2022
%F A002163 Equals w - w^2 - w^3 + w^4 where w = exp(2*Pi*i/5). - _Alexander R. Povolotsky_, Nov 23 2022
%F A002163 From _Antonio Graciá Llorente_, Apr 18 2024: (Start)
%F A002163 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)).
%F A002163 Equals Product_{k>=1} A217562(k)/A045572(k).
%F A002163 Equals Product_{k>=0} (1/2)*(((4*k + 9)/(4*k + 1))^(1/2) + ((4*k + 1)/(4*k + 9))^(1/2)).
%F A002163 Equals Product_{k>=1} (phi^k + phi)/(phi^k + phi - 1), with phi = A001622.
%F A002163 Equals Product_{k>=0} (Fibonacci(2*k + 3) + (-1)^k)/(Fibonacci(2*k + 3) - (-1)^k). (End)
%e A002163 2.236067977499789696409173668731276235440618359611525724270897245410520...
%t A002163 RealDigits[N[Sqrt[5],200]] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%o A002163 (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
%o A002163 (Magma) SetDefaultRealField(RealField(100)); Sqrt(5); // _Vincenzo Librandi_, Feb 13 2020
%Y A002163 Cf. A000032, A000045, A001622.
%Y A002163 Cf. A040002 (continued fraction).
%K A002163 nonn,cons
%O A002163 1,1
%A A002163 _N. J. A. Sloane_
%E A002163 Sequence corrected by _Paul Zimmermann_, Mar 15 1996
%E A002163 Additional comments from _Jason Earls_, Mar 26 2001