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%I A002193 M3195 N1291
%S A002193 1,4,1,4,2,1,3,5,6,2,3,7,3,0,9,5,0,4,8,8,0,1,6,8,8,7,2,4,2,0,9,6,9,
%T A002193 8,0,7,8,5,6,9,6,7,1,8,7,5,3,7,6,9,4,8,0,7,3,1,7,6,6,7,9,7,3,7,9,9,
%U A002193 0,7,3,2,4,7,8,4,6,2,1,0,7,0,3,8,8,5,0,3,8,7,5,3,4,3,2,7,6,4,1,5,7
%N A002193 Decimal expansion of square root of 2.
%C A002193 Sometimes called Pythagoras's constant.
%D A002193 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002193 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002193 S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and 
               its Applications, vol. 94, Cambridge University Press, Section 1.1.
%D A002193 M. Gardner, Gardner's Workout, Chapter 2 "The Square Root of 2=1.414213562373095..." 
               pp. 9-19 A. K. Peters MA 2002.
%D A002193 M. F. Jones, 22900D approximations to the square roots of the primes 
               less than 100, Math. Comp., 22 (1968), 234-235.
%D A002193 Uhler, Horace S.; Many-figures approximations to sqrt{2} and distribution 
               of digits in sqrt{2} and 1/sqrt{2}. Proc. Nat. Acad. Sci. U. S. A. 
               37, (1951). 63-67.
%D A002193 B. Rittaud, Le fabuleux destin de sqrt(2), Le Pommier, Paris 2006.
%D A002193 D. Flannery, The Square Root of 2, Copernicus Books Springer-Praxis Pub. 
               2006.
%H A002193 Harry J. Smith, <a href="b002193.txt">Table of n, a(n) for n=1,...,20000</
               a>
%H A002193 I. Khavkine, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               SquareRootOf2IsIrrationalProof.html">square root of 2 is irrational</
               a>
%H A002193 R. Nemiroff and J. Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/
               gifcity/sqrt2.1mil">The Square Root of Two to 1 Million Digits</a>
%H A002193 R. Nemiroff and J. Bonnell, <a href="http://www.ibiblio.org/pub/docs/
               books/gutenberg/etext94/2sqrt10a.txt">The Square Root of Two to 5 
               million digits</a>
%H A002193 R. Nemiroff and J. Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/
               gifcity/sqrt2.10mil">The first 10 million digits of the square root 
               of 2</a>
%H A002193 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               sqrt2.txt">The square root of 2 to 10 million digits</a>
%H A002193 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PythagorassConstant.html">Pythagoras's Constant</a>
%H A002193 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareRoot.html">Link to a section of The World of Mathematics.</
               a>
%H A002193 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PythagorassConstant.html">Link to a section of The World of Mathematics</
               a>
%H A002193 D. & J. Ensley, <a href="http://www.maa.org/reviews/roottwo.html">Review 
               of "The Square Root of 2" by D. Flannery</a>
%H A002193 H. S. Uhler, <a href="http://www.pnas.org/cgi/reprint/37/1/63.pdf">Many-Figure 
               Approximations To Sqrt(2), And Distribution Of Digits In Sqrt(2) 
               And 1/Sqrt(2)</a>
%H A002193 C. P. Simoes, <a href="http://www.cpsimoes.net/tdm/tdm.html">Teste de 
               Desempenho Mental</a>.
%F A002193 Sqrt(2) = 14 * sum_{n=0...infinity} (A001790(n)/2^A005187(floor(n/2)) 
               * 10^(-2n-1)) where A001790(n) are numerators in expansion of 1/sqrt(1-x) 
               and the denominators in expansion of 1/sqrt(1-x) are 2^A005187(n). 
               14 = 2*7, see A010503 (expansion of 1/sqrt(2)). - Gerald McGarvey 
               (Gerald.McGarvey(AT)comcast.net), Jan 01 2005
%F A002193 Limit(n-->+oo) of (1/n)*(sum(k=1, n, frac(sqrt(1+zeta(k+1))))) = 1/(1+sqrt(2)) 
               - Yalcin Aktar (aktaryalcin(AT)msn.com), Jul 14 2005
%F A002193 sqrt(2)=2+n*A167199(n-1)/A167199(n) as n-->infinity (conjecture). [From 
               Mats Granvik (mats.granvik(AT)abo.fi), Oct 30 2009]
%e A002193 sqrt(2) = 1.41421356237309504880168872420969807856967187537694807317667... 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
%t A002193 RealDigits[N[2^(1/2),6! ]] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Dec 25 2008]
%o A002193 (PARI) { default(realprecision, 20080); x=sqrt(2); for (n=1, 20000, d=floor(x); 
               x=(x-d)*10; write("b002193.txt", n, " ", d)); } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
%o A002193 Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Oct 
               20 2009: (Start)
%o A002193 (PARI) r=0; x=2; /* Digit-by-digit method */
%o A002193 for(digits=1,100,{d=0;while((20*r+d)*d <= x,d++);
%o A002193 d--; /* while loop overshoots correct digit */
%o A002193 print(d);x=100*(x-(20*r+d)*d);r=10*r+d}) (End)
%Y A002193 Cf. A020807.
%Y A002193 Cf. A010503, A001790, A005187.
%Y A002193 Sequence in context: A050338 A077088 A156896 this_sequence A020807 A055190 
               A155781
%Y A002193 Adjacent sequences: A002190 A002191 A002192 this_sequence A002194 A002195 
               A002196
%K A002193 nonn,cons
%O A002193 1,2
%A A002193 N. J. A. Sloane (njas(AT)research.att.com).
%E A002193 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009

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