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A004539 Expansion of sqrt(2) in base 2. 10
1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

B. Adamczewski and N. Rampersad, On patterns occurring in binary algebraic numbers, Proc. Amer. Math. Soc. 136 (2008), 3105-3109.

D. Bailey et al., On the binary expansions of algebraic numbers, J. Théor. Nombres Bordeaux, 16 (2004), 487-518.

R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. Zbl 201.04705.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Richard Isaac, On the simple normality to base 2 of the square root of s, for s not a perfect square., arXiv:math/0512404 [math.NT], 2005-2006.

Mariusz Iwaniuk, Formula for computing sqrt(2) of binary numbers

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

Thomas Stoll, On a problem of Erdős and Graham concerning digits, Acta Arithmetica 125(2006), 89-100.

Thomas Stoll, A fancy way to obtain the binary digits of 759250125 sqrt{2}, (2009),  Amer. Math. Monthly, 117 (2010), 611-617.

Eric Weisstein's World of Mathematics, Wolfram's Iteration

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

FORMULA

a(n) = 1/2 - (2*ArcTan(Cot(2^(-(3/2)+n)*Pi)))/Pi + ArcTan(Cot(2^(-(1/2)+n)*Pi))/Pi for n > 0 and n in Z. See link for a proof. - Mariusz Iwaniuk, Apr 20 2017

a(n) = floor(2^(-(1/2) + n)) - 2*floor(2^(-(3/2) + n)) for n > 0 and n in Z. See link for a proof. - Mariusz Iwaniuk, Apr 26 2017

EXAMPLE

1.0110101000001001111001...

MATHEMATICA

N[Sqrt[2], 200]; RealDigits[%, 2]

PROG

(bc) obase=2 scale=200 sqrt(2)

(Haskell)

a004539 n = a004539_list !! (n-1)

a004539_list = w 2 0 where

   w x r = bit : w (4 * (x - (4 * r + bit) * bit)) (2 * r + bit)

     where bit = head (dropWhile (\b -> (4 * r + b) * b < x) [0..]) - 1

-- Reinhard Zumkeller, Dec 16 2013

CROSSREFS

Cf. A002193 (decimal version), A233836 (run lengths of 0s and 1s).

Sequence in context: A129360 A129372 A169591 * A023960 A129686 A104974

Adjacent sequences:  A004536 A004537 A004538 * A004540 A004541 A004542

KEYWORD

nonn,base,cons

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 17 13:03 EDT 2017. Contains 290635 sequences.