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) = 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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved