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A002193 Decimal expansion of square root of 2.
(Formerly M3195 N1291)
174
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sometimes called Pythagoras's constant.

Its continued fraction expansion is [1; 2, 2, 2, ...] (see A040000). - Arkadiusz Wesolowski, Mar 10 2012

REFERENCES

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1.

David Flannery, The Square Root of 2, Copernicus Books Springer-Praxis Pub. 2006.

Martin Gardner, Gardner's Workout, Chapter 2 "The Square Root of 2=1.414213562373095..." pp. 9-19 A. K. Peters MA 2002.

B. Rittaud, Le fabuleux destin de sqrt(2), Le Pommier, Paris 2006.

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).

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000

D. & J. Ensley, Review of "The Square Root of 2" by D. Flannery

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62, Hermite's constant gamma_4.

M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., 22 (1968), 234-235.

I. Khavkine, PlanetMath.org, square root of 2 is irrational

Jason Kimberley, Index of expansions of sqrt(d) in base b

Robert Nemiroff and Jerry Bonnell, The Square Root of Two to 1 Million Digits

Robert Nemiroff and Jerry Bonnell, The Square Root of Two to 5 million digits

Robert Nemiroff and Jerry Bonnell, The first 10 million digits of the square root of 2

_Simon Plouffe_, Plouffe's Inverter, The square root of 2 to 10 million digits

Simon Plouffe, Generalized expansion of real constants

M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013.

Carlos P. Simoes, Mental performance test, part I item 13.

Horace S. Uhler, Many-Figure 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.

Eric Weisstein's World of Mathematics, Pythagoras's Constant

Eric Weisstein's World of Mathematics, Square Root

FORMULA

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, Jan 01 2005

Limit(n --> +infinity) of (1/n)*(sum(k = 1, n, frac(sqrt(1+zeta(k+1))))) = 1/(1+sqrt(2)). - Yalcin Aktar, Jul 14 2005

sqrt(2) = 2 + n*A167199(n-1)/A167199(n) as n --> infinity (conjecture). [Mats Granvik, Oct 30 2009]

sqrt(2) = limit as n goes to infinity of A179807(n+1)/A179807(n) - 1. - Mats Granvik, Feb 15 2011

From Wolfdieter Lang, Oct 22 2013: (Start)

sqrt(2) = product(2*cos((2*l+1)*Pi/(4*k)), l = 0..k-1) = product(R(2*l+1,rho(4*k)), l = 0..k - 1), identical for k >= 1, with the row polynomials R(n, x) from A127672 and rho(4*k) := 2*cos(Pi/(4*k)) is the length ratio (smallest diagonal)/side in a regular (4*k)-gon. From the product formula given in a Oct 21 2013 formula contribution to A056594, with n -> 2*k, using cos(Pi-alpha) = - cos(alpha) to obtain 2 for the square of the present product. (End)

EXAMPLE

1.41421356237309504880168872420969807856967187537694807317667...

MAPLE

Digits:=100; evalf(sqrt(2)); # Wesley Ivan Hurt, Dec 04 2013

MATHEMATICA

RealDigits[N[2^(1/2), 128]] (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)

PROG

(PARI) { default(realprecision, 20080); x=sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002193.txt", n, " ", d)); } \\ Harry J. Smith, Apr 21 2009

(PARI) r=0; x=2; /* Digit-by-digit method */

for(digits=1, 100, {d=0; while((20*r+d)*d <= x, d++);

d--; /* while loop overshoots correct digit */

print(d); x=100*(x-(20*r+d)*d); r=10*r+d}) // Michael B. Porter, Oct 20 2009

(Maxima) fpprec: 100$ ev(bfloat(sqrt(2))); /* Martin Ettl, Oct 17 2012 */

(Haskell)  -- After Michael B. Porter's PARI program.

a002193 n = a002193_list !! (n-1)

a002193_list = w 2 0 where

   w x r = dig : w (100 * (x - (20 * r + dig) * dig)) (10 * r + dig)

     where dig = head (dropWhile (\d -> (20 * r + d) * d < x) [0..]) - 1

-- Reinhard Zumkeller, Nov 22 2013

CROSSREFS

Cf. A020807, A010503, A001790, A005187.

Cf. A004539 (binary version).

Sequence in context: A077088 A156896 * A020807 A188582 A230077 A055190

Adjacent sequences:  A002190 A002191 A002192 * A002194 A002195 A002196

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 25 16:38 EST 2014. Contains 249998 sequences.