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

0,2

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

Formula

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

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

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 *)

Prog

(PARI) a(n)=floor(sqrt(2)<<n) \\ Charles R Greathouse IV, Sep 22 2011
(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
(PARI) {a(n) = sqrtint(2*4^n)}; /* Michael Somos, Oct 29 2016 */

Crossrefs

Cf. A084185, A084186, A017910.

Sequence in context: A071015 A293362 A362583 * A266721 A044432 A033120

Adjacent sequences: A084185 A084186 A084187 * A084189 A084190 A084191

Keyword

nonn,easy

Author

Ralf Stephan, May 18 2003

Status

approved