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A010467 Decimal expansion of square root of 10.

%I A010467 #99 Jul 30 2025 17:48:11
%S A010467 3,1,6,2,2,7,7,6,6,0,1,6,8,3,7,9,3,3,1,9,9,8,8,9,3,5,4,4,4,3,2,7,1,8,
%T A010467 5,3,3,7,1,9,5,5,5,1,3,9,3,2,5,2,1,6,8,2,6,8,5,7,5,0,4,8,5,2,7,9,2,5,
%U A010467 9,4,4,3,8,6,3,9,2,3,8,2,2,1,3,4,4,2,4,8,1,0,8,3,7,9,3,0,0,2,9
%N A010467 Decimal expansion of square root of 10.
%C A010467 Continued fraction expansion is 3 followed by {6} repeated. - _Harry J. Smith_, Jun 02 2009
%C A010467 In 1594, Joseph Scaliger claimed Pi = sqrt(10), but Ludolph van Ceulen immediately knew this to be wrong. - _Alonso del Arte_, Jan 17 2013
%C A010467 The 7th-century Hindu mathematician Brahmagupta used this constant as value of Pi. - _Stefano Spezia_, Jul 08 2022
%D A010467 Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.
%D A010467 Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 112.
%D A010467 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 238.
%D A010467 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.
%D A010467 Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, pp. 89, 91.
%D A010467 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 55.
%H A010467 Harry J. Smith, <a href="/A010467/b010467.txt">Table of n, a(n) for n = 1..20000</a>
%H A010467 Josep M. Brunat and Joan-Carles Lario, <a href="https://arxiv.org/abs/2103.05306">A problem on concatenated integers</a>, arXiv:2103.05306 [math.NT], 2021. For 1/sqrt(10).
%H A010467 Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/sqrt_base">Index of expansions of sqrt(d) in base b</a>.
%H A010467 Robert Nemiroff and Jerry Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt10.1mil">The first 1 million digits of square root of 10</a>.
%H A010467 Robert Nemiroff and Jerry Bonnell, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/sqrt10.txt">The first 1 million digits of square root of 10</a>.
%H A010467 J. J. O'Connor and E. F. Robertson, <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Van_Ceulen/">Ludolph Van Ceulen</a>.
%H A010467 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F A010467 Sqrt(10) = sqrt(1 + i*sqrt(15)) + sqrt(1 - i*sqrt(15)) = sqrt(1/2 + 2*i*sqrt(5)) + sqrt(1/2 - 2*i*sqrt(5)), where i = sqrt(-1). - _Bruno Berselli_, Nov 20 2012
%F A010467 Equals 1/sqrt(10), with offset 0. - _Michel Marcus_, Mar 10 2021
%F A010467 Equals 2 + Sum_{k>=1} Lucas(k)*binomial(2*k,k)/8^k. - _Amiram Eldar_, Jan 17 2022
%F A010467 a(k) = floor(Sum_{n>=1} A005875(n)/exp(Pi*n/(10^((2/3)*k+(1/3))))) mod 10. Will give the k-th decimal digit of sqrt(10). A005875 : number of ways to write n as sum of 3 squares. - _Simon Plouffe_, Dec 30 2023
%e A010467 3.162277660168379331998893544432718533719555139325216826857504852792594...
%t A010467 RealDigits[N[Sqrt[10],200]] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%o A010467 (PARI)  default(realprecision, 20080); x=sqrt(10); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010467.txt", n, " ", d));  \\ _Harry J. Smith_, Jun 02 2009
%o A010467 (Magma) SetDefaultRealField(RealField(100)); Sqrt(10); // _Vincenzo Librandi_, Feb 15 2020
%Y A010467 Cf. A000032, A040006 (continued fraction).
%K A010467 nonn,cons
%O A010467 1,1
%A A010467 _N. J. A. Sloane_