A084188 a(0)=1, a(n+1) = 2*a(n) + b(n+2), where b(n)=A004539(n) is the n-th bit in the binary expansion of sqrt(2).

1, 2, 5, 11, 22, 45, 90, 181, 362, 724, 1448, 2896, 5792, 11585, 23170, 46340, 92681, 185363, 370727, 741455, 1482910, 2965820, 5931641, 11863283, 23726566, 47453132, 94906265, 189812531, 379625062, 759250124

Offset: 0

Author

Ralf Stephan, May 18 2003

Keywords

• nonn
• easy

Comments

Numerators in approximation sqrt(2) ~ a(n)/2^n.

a(n) is the number k such that {log_2(k)} < 1/2 < {log_2(k+1)}, where { } = fractional part. Equivalently, the jump sequence of f(x) = log_2(x), in the sense that these are the positive integers k for which round(log_2(k)) < round(log_2(k+1)); see A219085. - Clark Kimberling, Jan 01 2013

Largest k such that k^2 <= 2^(2n + 1). - Irina Gerasimova, Jul 07 2013

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A084185, A084186, A017910.

Programs

Haskell

a084188 n = a084188_list !! n
a084188_list = scanl1 (\u v -> 2 * u + v) a004539_list
-- Reinhard Zumkeller, Dec 16 2013

Magma

[Isqrt(2^(2*n+1)):n in[0..40]]; // Jason Kimberley, Oct 25 2016

Maple

A084188 := n->floor(sqrt(2)*2^n); # Peter Luschny, Sep 20 2011

Mathematica

Table[Floor[Sqrt[2] 2^n],{n,0,30}] (* Harvey P. Dale, Aug 15 2013 *)

PARI

a(n)=floor(sqrt(2)<Charles R Greathouse IV, Sep 22 2011

PARI

{a(n) = sqrtint(2*4^n)}; /* Michael Somos, Oct 29 2016 */

Python

from math import isqrt
def A084188(n): return isqrt(1<<(n<<1)+1) # Chai Wah Wu, Jan 24 2024

Formula

a(n) = floor(sqrt(2)*2^n).

a(n) = A017910(2*n+1). - Peter Luschny, Sep 20 2011