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

1,1

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

References

I. J. Good and T. N. Gover, The generalized serial test and the binary expansion of sqrt(2), J. Royal Statistical Society, Series A, Vol. 130, Part 1, 1967, pp. 102-107.

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

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

Example

1.0110101000001001111001...

Mathematica

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

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
(PARI) binary(sqrt(2)) \\ Michel Marcus, Nov 06 2017
(PARI) a(n) = floor(quadgen(8)<<(n-1))%2; \\ Chittaranjan Pardeshi, Sep 09 2024

Crossrefs

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

Sequence in context: A169591 A374646 A189295 * A354805 A294878 A023960

Adjacent sequences: A004536 A004537 A004538 * A004540 A004541 A004542

Keyword

nonn,base,cons

Author

N. J. A. Sloane

Status

approved