A004539 Expansion of sqrt(2) in base 2.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• cons

Comments

Bailey, Borwein, Crandall, & Pomerance prove a general result that the first n terms contain >> sqrt(n) 1's. Vandehey improves this to sqrt(2*n)(1 + o(1)). - Charles R Greathouse IV, Nov 07 2017

Examples

			1.0110101000001001111001...
		

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.
• David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, 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.
• Jason Kimberley, Index of expansions of sqrt(d) in base b
• Thomas Stoll, On families of nonlinear recurrences related to digits, Journal of Integer Sequences 8 (2005), 05.3.2.
• Thomas Stoll, On a problem of Erdős and Graham concerning digits, Acta Arithmetica 125 (2006), pp. 89-100.
• Thomas Stoll, A fancy way to obtain the binary digits of 759250125 sqrt{2}, (2009), Amer. Math. Monthly, 117 (2010), 611-617.
• Joseph Vandehey, On the binary digits of sqrt(2), arXiv:1711.01722 [math.NT], 2017.
• Eric Weisstein's World of Mathematics, Wolfram's Iteration
• Eric Weisstein's World of Mathematics, Pythagoras's Constant

Crossrefs

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

Programs

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

Mathematica

N[Sqrt[2], 200]; RealDigits[%, 2]
RealDigits[Sqrt[2],2,120][[1]] (* Harvey P. Dale, Aug 03 2024 *)

PARI

binary(sqrt(2)) \\ Michel Marcus, Nov 06 2017

PARI

a(n) = floor(quadgen(8)<<(n-1))%2; \\ Chittaranjan Pardeshi, Sep 09 2024

bc

obase=2 scale=200 sqrt(2)

Formula

a(k) = floor(Sum_{n>=1} A005875(n)/exp(Pi*n/(2^((2/3)*k+(1/3))))) mod 2. Will give the k-th binary digit of sqrt(2). A005875 : number of ways to write n as sum of 3 squares. - Simon Plouffe, Dec 30 2023