A017910 Powers of sqrt(2) rounded down.

1, 1, 2, 2, 4, 5, 8, 11, 16, 22, 32, 45, 64, 90, 128, 181, 256, 362, 512, 724, 1024, 1448, 2048, 2896, 4096, 5792, 8192, 11585, 16384, 23170, 32768, 46340, 65536, 92681, 131072, 185363, 262144, 370727, 524288

Offset

0,3

Comments

a(n) is the number of positive squares <= 2^n (cf. A136417). - Hans Havermann, Apr 05 2008

If expressed to two significant digits, these are the f-stop numbers in photography: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, ...

There are also "half stops" (sqrt(2)^(n/2)) and "third stops" (sqrt(2)^(n/3)): 1, 1.4, 1.6, 1.8, 2.0, 2.2, 2.5, 2.8, 3.2, 3.6, 4, 4.5, 5, 5.7, 6.3, 7.1, 8, 9, 10.

a(n) is also the ratio (rounded down) of the curvature of the circle inscribed in the n-th 45-45-90 triangle to that of the circle inscribed in the 1st triangle, with the triangles arranged in a spiral as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 28 2013

a(n) is also the total length of Heighway dragon (rounded down) after n-iterations when L(0) = 1. See illustration in links. - Kival Ngaokrajang, Dec 15 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Kival Ngaokrajang, Illustration of initial terms.

Kival Ngaokrajang, Illustration of Heighway dragon for n = 0..5.

Wikipedia, Dragon curve.

Formula

a(n) = A000196(A000079(n)). - Jason Kimberley, Oct 28 2016

a(n) = A017912(n)-1 if n is odd. a(n) = A017912(n) = 2^(n/2) if n is even. - Chai Wah Wu, Jul 26 2022

Maple

A017910 := n->floor(sqrt(2^n)); # Peter Luschny, Sep 20 2011

Mathematica

Floor[(Sqrt[2])^Range[0, 40]] (* Vincenzo Librandi, Nov 20 2011 *)

Prog

(PARI) a(n)=sqrtint(2^n) \\ Charles R Greathouse IV, Sep 22 2011
(Magma) [Floor(Sqrt(2^n)): n in [0..40]]; // Vincenzo Librandi, Nov 20 2011
(Magma) [Isqrt(2^n):n in[0..40]]; // Jason Kimberley, Oct 25 2016
(Python)
from math import isqrt
def A017910(n): return isqrt(1<<n) if n&1 else 1<<(n>>1) # Chai Wah Wu, Jul 26 2022

Crossrefs

Cf. A136417, A017912. Bisections: A000079, A084188.

Partial sums of A190568.

Sequence in context: A109434 A371840 A089299 * A240734 A328460 A238478

Adjacent sequences: A017907 A017908 A017909 * A017911 A017912 A017913

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved