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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

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

Partial sums of A190568.

Programs

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

Maple

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

Mathematica

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

PARI

a(n)=sqrtint(2^n) \\ Charles R Greathouse IV, Sep 22 2011

Python

from math import isqrt
def A017910(n): return isqrt(1<>1) # Chai Wah Wu, Jul 26 2022

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